A118841
Numbers n such that Ceiling[GoldenRatio^n] is prime.
http://www.research.att.com/~njas/sequences/A118841
1, 2, 3, 4, 8, 16
COMMENT
a(7)>3000 or non-existent. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 07 2007
==========================================================
With all the excellent mathematicians and physicists on this board, and as trivial as this question may be, this ought to be a piece of cake...
COMPANION THREAD
Highly Composite Number "divisor Threads", Can anyone explain this?
http://www.physforum.com/index.php?showtopic=25332
Best,
Raphie
P.S. There is a moral to this story, and I will make it very simple. NOONE will be able to answer this question. Of this I am quite certain. Point being, in no way, shape or form do we have it all figured out, and anyone who thinks otherwise is a fool. And anyone who judges, much less censors, those exploring new methods and means to acquire knowledge is also far less than wise. - RF
Bukh said it best IMHO.
QUOTE (bukh+Feb 16 2009, 12:12 AM)
I stumbled over these Posting Guidelines put forward in another Discussion Forum. And I am not going to revealing which one.
"Sciences
There will be no discussions of highly speculative subjects in the Science Forums.
Topics or responses dealing with the pixel universe, the æther, or other perennially resurgent pseudo-science will be deleted.
If a member abuses these guidelines their posting permissions for this forum will be removed, either for a probationary period or permanently, at the administrator's discretion."
Let us pray to that Mother Nature follow Scientific rules and regulations.
"Sciences
There will be no discussions of highly speculative subjects in the Science Forums.
Topics or responses dealing with the pixel universe, the æther, or other perennially resurgent pseudo-science will be deleted.
If a member abuses these guidelines their posting permissions for this forum will be removed, either for a probationary period or permanently, at the administrator's discretion."
Let us pray to that Mother Nature follow Scientific rules and regulations.
QUOTE (Raphie Frank+Mar 18 2009, 04:47 AM)
With all the excellent mathematicians and physicists on this board, and as trivial as this question may be, this ought to be a piece of cake...
OH MY GOD! You have gone alphanumeric on us!
Seriously, R., you didn't ask any question! ! !
OH MY GOD! You have gone alphanumeric on us!
Seriously, R., you didn't ask any question! ! !
Note: In response to a private email, in this specific instance:
CEILING --> Closest Integer above
FLOOR --> Closest Integer below
The output looks something like this:
CEILING --> Closest Integer above
FLOOR --> Closest Integer below
QUOTE (IAMoraes+Mar 19 2009, 01:53 AM)
OH MY GOD! You have gone alphanumeric on us!
Seriously, R., you didn't ask any question! ! !
No question?
"N Such That Ceiling[goldenratio^n] Is Prime, Can anyone explain this?"
A tip: The question mark indicates when a question has been asked. :-)
But, seriously, in anthropomorphized terms, the Ceiling phi^n series seems to "know" how "not to be prime" past [Ceiling phi^16]. 1/2 of those numbers are even, so that's easy to explain, but of the other 1492 or so numbers [(3000 - 16)/2] that have been tested for not even 1 to be prime is suggestive of some phenomeneon at work.
Conversely, you should know, it is not uncommon for Lucas numbers themselves to be prime. Here is a number sequence from OEIS:
=============================================================
A005479 Prime Lucas numbers (cf. A000032)
http://www.research.att.com/~njas/sequences/A005479
2, 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688846502588399, 32361122672259149, 412670427844921037470771
=============================================================
The indices for those Lucas numbers up above, past 16 (L_16 = 2207), it so happens, are all prime numbers (although I am not sure if this always holds) and the only exceptions are phi^1, phi^4, phi^8, phi^16, the exponents of which are all powers of 2. (Ceiling phi^2 is also prime)
More specifically, the question is two-fold:
A ) Why is it that the series of integers "n Such That Ceiling[goldenratio^n] Is Prime" "seems to" ends at 16?
B ) Why is it that the series of integers "n Such That Ceiling[goldenratio^n] Is Prime" includes the powers of 2 from 2^0 (1) through 2^4 (16) but not thereafter?
All this said, Ivan, this post is an exercise in futility. Noone will answer the question because noone, I dare say, is capable of answering it. And quite frankly, I am very surprised that I have not yet been called an "idiot" or some other name simply for asking the question (indicated by a "?")
Best,
Raphie
And far more intuitively speaking, is it not possible to add in a C) and and ask:
C) Is this phenomeneon not perhaps related in some tangential manner to the BBP formula for pi, whuch is based on Base 16?
===========================================================
Finding the N-th digit of Pi
Here is a very interesting formula for pi, discovered by David Bailey, Peter Borwein, and Simon Plouffe in 1995:
Pi = SUMk=0 to infinity 16-k [ 4/(8k+1) - 2/(8k+4) - 1/(8k+5) - 1/(8k+6) ].
The reason this pi formula is so interesting is because it can be used to calculate the N-th digit of Pi (in base 16) without having to calculate all of the previous digits!
Su, Francis E., et al. "Finding the N-th digit of Pi." Mudd Math Fun Facts. <http://www.math.hmc.edu/funfacts>.
also see:
BBP Formula
http://mathworld.wolfram.com/BBPFormula.html
===========================================================
Seriously, R., you didn't ask any question! ! !
No question?
"N Such That Ceiling[goldenratio^n] Is Prime, Can anyone explain this?"
A tip: The question mark indicates when a question has been asked. :-)
But, seriously, in anthropomorphized terms, the Ceiling phi^n series seems to "know" how "not to be prime" past [Ceiling phi^16]. 1/2 of those numbers are even, so that's easy to explain, but of the other 1492 or so numbers [(3000 - 16)/2] that have been tested for not even 1 to be prime is suggestive of some phenomeneon at work.
Conversely, you should know, it is not uncommon for Lucas numbers themselves to be prime. Here is a number sequence from OEIS:
=============================================================
A005479 Prime Lucas numbers (cf. A000032)
http://www.research.att.com/~njas/sequences/A005479
2, 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688846502588399, 32361122672259149, 412670427844921037470771
=============================================================
The indices for those Lucas numbers up above, past 16 (L_16 = 2207), it so happens, are all prime numbers (although I am not sure if this always holds) and the only exceptions are phi^1, phi^4, phi^8, phi^16, the exponents of which are all powers of 2. (Ceiling phi^2 is also prime)
More specifically, the question is two-fold:
A ) Why is it that the series of integers "n Such That Ceiling[goldenratio^n] Is Prime" "seems to" ends at 16?
B ) Why is it that the series of integers "n Such That Ceiling[goldenratio^n] Is Prime" includes the powers of 2 from 2^0 (1) through 2^4 (16) but not thereafter?
All this said, Ivan, this post is an exercise in futility. Noone will answer the question because noone, I dare say, is capable of answering it. And quite frankly, I am very surprised that I have not yet been called an "idiot" or some other name simply for asking the question (indicated by a "?")
Best,
Raphie
And far more intuitively speaking, is it not possible to add in a C) and and ask:
C) Is this phenomeneon not perhaps related in some tangential manner to the BBP formula for pi, whuch is based on Base 16?
===========================================================
Finding the N-th digit of Pi
Here is a very interesting formula for pi, discovered by David Bailey, Peter Borwein, and Simon Plouffe in 1995:
Pi = SUMk=0 to infinity 16-k [ 4/(8k+1) - 2/(8k+4) - 1/(8k+5) - 1/(8k+6) ].
The reason this pi formula is so interesting is because it can be used to calculate the N-th digit of Pi (in base 16) without having to calculate all of the previous digits!
Su, Francis E., et al. "Finding the N-th digit of Pi." Mudd Math Fun Facts. <http://www.math.hmc.edu/funfacts>.
also see:
BBP Formula
http://mathworld.wolfram.com/BBPFormula.html
===========================================================
QUOTE (Raphie Frank+Mar 19 2009, 11:48 AM)
Noone will answer the question because noone, I dare say, is capable of answering it
Oh, yeah?
Oh, yeah?
Oh, yeah?
I am going to go learn math RIGHT NOW just to prove you wrong!
Oh, yeah?
Oh, yeah?
Oh, yeah?
I am going to go learn math RIGHT NOW just to prove you wrong!
a(n)
= Ceiling( (½ + ½√5)ⁿ )
= ½ + (½ + ½√5)ⁿ + (½ − ½√5)ⁿ − ½ (−1)ⁿ
= ½ + (½ + ½√5)ⁿ + (−2/(1 + √5))ⁿ − ½ (−1)ⁿ
= a(n−1) + 2 a(n−1) − a(n−2) − a(n−3) ( for a(1..4) = 2, 3, 5, 7 )
is a fast-growing (approximately exponential) integer sequence. But this last formula is useful for testing primality of large members of the sequence on toy calculators (like Excel).
Modulo 2 it is periodic of period 6
Modulo 3 it is periodic of period 8
Modulo 5 it is periodic of period 4
Modulo 7 it is periodic of period 16
Modulo 47 it is periodic of period 32
Modulo 2207 it is periodic of period 64
Therefore modulo 2×3×5×7×47×2207=21783090 it is periodic of period 3×64=192
a(192k + 32) ≡ 4870847 mod 21783090
a(192k + 64) ≡ 10891547 mod 21783090
a(192k + 128) ≡ 10891547 mod 21783090
a(192k + 129) ≡ 10891547 mod 21783090
a(192k + 160) ≡ 4870847 mod 21783090
So past a(16) only 5/192 of the series are even candidates for prime numbers.
But primes get increasingly rare as you go to higher numbers.
a(32) = 1087×4481 -- both factors 1087 and 4481 have periods of 128 in this sequence. So a(2k×192+32) also cannot be prime.
So only 9/384 of the sequence past a(16) remain as prime candidates. The first candidate which has not been excluded is a(64) = 23725150497407 = 127×186812208641
Factor 127 has periodicity 256
a(256k), a(256k + 64), a(256k + 129), a(256k + 192) are multiples of 127.
a(768k + 32) ≡ 0 mod 1087
a(768k + 64) ≡ 0 mod 127
a(768k + 128) ≡ 10891547 mod 21783090
a(768k + 129) ≡ 0 mod 127
a(768k + 160) ≡ 0 mod 127
a(768k + 224) ≡ 0 mod 1087
a(768k + 256) ≡ 10891547 mod 21783090
a(768k + 320) ≡ 0 mod 127
a(768k + 321) ≡ 0 mod 1087
a(768k + 352) ≡ 0 mod 1087
a(768k + 416) ≡ 0 mod 1087
a(768k + 448) ≡ 0 mod 127
a(768k + 512) ≡ 10891547 mod 21783090
a(768k + 513) ≡ 10891547 mod 21783090
a(768k + 544) ≡ 0 mod 1087
a(768k + 608) ≡ 0 mod 1087
a(768k + 640) ≡ 10891547 mod 21783090
a(768k + 704) ≡ 0 mod 127
a(768k + 705) ≡ 0 mod 1087
a(768k + 736) ≡ 0 mod 1087
So even though these numbers are very large, just a handful of primes:
2, 3, 5, 7, 47, 127, 1087 and 2207 factor all but 5/768 of the sequence past a(16).
It is left as an exercise for the reader to find factors of the series for 128, 256, 512, 513, 640, 896, 1024, 1280, 1281, 1408, 1664, 1792, 2048, 2049, 2176, 2432, 2560, 2816, 2817, 2944 -- and an open problem to establish any new periodicities.
= Ceiling( (½ + ½√5)ⁿ )
= ½ + (½ + ½√5)ⁿ + (½ − ½√5)ⁿ − ½ (−1)ⁿ
= ½ + (½ + ½√5)ⁿ + (−2/(1 + √5))ⁿ − ½ (−1)ⁿ
= a(n−1) + 2 a(n−1) − a(n−2) − a(n−3) ( for a(1..4) = 2, 3, 5, 7 )
is a fast-growing (approximately exponential) integer sequence. But this last formula is useful for testing primality of large members of the sequence on toy calculators (like Excel).
Modulo 2 it is periodic of period 6
Modulo 3 it is periodic of period 8
Modulo 5 it is periodic of period 4
Modulo 7 it is periodic of period 16
Modulo 47 it is periodic of period 32
Modulo 2207 it is periodic of period 64
Therefore modulo 2×3×5×7×47×2207=21783090 it is periodic of period 3×64=192
a(192k + 32) ≡ 4870847 mod 21783090
a(192k + 64) ≡ 10891547 mod 21783090
a(192k + 128) ≡ 10891547 mod 21783090
a(192k + 129) ≡ 10891547 mod 21783090
a(192k + 160) ≡ 4870847 mod 21783090
So past a(16) only 5/192 of the series are even candidates for prime numbers.
But primes get increasingly rare as you go to higher numbers.
a(32) = 1087×4481 -- both factors 1087 and 4481 have periods of 128 in this sequence. So a(2k×192+32) also cannot be prime.
So only 9/384 of the sequence past a(16) remain as prime candidates. The first candidate which has not been excluded is a(64) = 23725150497407 = 127×186812208641
Factor 127 has periodicity 256
a(256k), a(256k + 64), a(256k + 129), a(256k + 192) are multiples of 127.
a(768k + 32) ≡ 0 mod 1087
a(768k + 64) ≡ 0 mod 127
a(768k + 128) ≡ 10891547 mod 21783090
a(768k + 129) ≡ 0 mod 127
a(768k + 160) ≡ 0 mod 127
a(768k + 224) ≡ 0 mod 1087
a(768k + 256) ≡ 10891547 mod 21783090
a(768k + 320) ≡ 0 mod 127
a(768k + 321) ≡ 0 mod 1087
a(768k + 352) ≡ 0 mod 1087
a(768k + 416) ≡ 0 mod 1087
a(768k + 448) ≡ 0 mod 127
a(768k + 512) ≡ 10891547 mod 21783090
a(768k + 513) ≡ 10891547 mod 21783090
a(768k + 544) ≡ 0 mod 1087
a(768k + 608) ≡ 0 mod 1087
a(768k + 640) ≡ 10891547 mod 21783090
a(768k + 704) ≡ 0 mod 127
a(768k + 705) ≡ 0 mod 1087
a(768k + 736) ≡ 0 mod 1087
So even though these numbers are very large, just a handful of primes:
2, 3, 5, 7, 47, 127, 1087 and 2207 factor all but 5/768 of the sequence past a(16).
It is left as an exercise for the reader to find factors of the series for 128, 256, 512, 513, 640, 896, 1024, 1280, 1281, 1408, 1664, 1792, 2048, 2049, 2176, 2432, 2560, 2816, 2817, 2944 -- and an open problem to establish any new periodicities.
Well, I found two more useful prime factors, and the numbers between a(16) and a(3000) which haven't been sieved out already are:
a(512) = 100385689891921376688754239992826256704879627683181901515099398613465618884806971304035121947368905594088447
a(1024) = 10077286735077005660982008061065073068074475300466012444629388487574769652115651763500026128367679301744790365920278775601766000217455997930809875108639504578766853603625505162682177708433023235042368022152858871807
a(2048) = 101551707940958976476873723686691537560124936216049957128359991876726343954548826671188589246207404049249584475544225686479598181242824239865026718975582831790339042655215629893459077413144429144815676624628680986839974189561981371422914234508007983748918604313893297542075331365600286467195288043916251150089236097377941252734677327147814393642121714369673795261602383375518218563568418129690432349780011400594146198108859445247
a(2049) = 164314115064074223864576148000612998847585174640366089287959555695337458290006499227703037573972995285454985789647370334838861606641007166925922878842331205382589695052501971427380762698723623283313307472171427555245011577445371125109567321051020154831079307706336046259951456887961686612382210782486918934051933025188246199559687939504110732095847643418792008093685986825684713516873234428712977018731540771144530783255086432877
a(2560) = 10194338261356077326160187799937652767877792122376047993613531201778011435316238581095249774475993494792883712985604835086819811370688354738136993969473704668335140999009344672760777252421138274632160919163922548578686311643310500810215459723143805269213399447150542322544732417598852574256637198462963153786317294618477667041891210515901774780293384019411335280882067893169184308205553775137932511813897934335263447378300929559613107719465053344940923125698272637332584902371411782946761303134397195657180991572207443365373111673136127
And then using high powered tools (PHP if you can believe it) I verified no more primes up to a(51201) But a(1024) is composite with smallest factor of more than 2 billion.
a(512) = 100385689891921376688754239992826256704879627683181901515099398613465618884806971304035121947368905594088447
a(1024) = 10077286735077005660982008061065073068074475300466012444629388487574769652115651763500026128367679301744790365920278775601766000217455997930809875108639504578766853603625505162682177708433023235042368022152858871807
a(2048) = 101551707940958976476873723686691537560124936216049957128359991876726343954548826671188589246207404049249584475544225686479598181242824239865026718975582831790339042655215629893459077413144429144815676624628680986839974189561981371422914234508007983748918604313893297542075331365600286467195288043916251150089236097377941252734677327147814393642121714369673795261602383375518218563568418129690432349780011400594146198108859445247
a(2049) = 164314115064074223864576148000612998847585174640366089287959555695337458290006499227703037573972995285454985789647370334838861606641007166925922878842331205382589695052501971427380762698723623283313307472171427555245011577445371125109567321051020154831079307706336046259951456887961686612382210782486918934051933025188246199559687939504110732095847643418792008093685986825684713516873234428712977018731540771144530783255086432877
a(2560) = 10194338261356077326160187799937652767877792122376047993613531201778011435316238581095249774475993494792883712985604835086819811370688354738136993969473704668335140999009344672760777252421138274632160919163922548578686311643310500810215459723143805269213399447150542322544732417598852574256637198462963153786317294618477667041891210515901774780293384019411335280882067893169184308205553775137932511813897934335263447378300929559613107719465053344940923125698272637332584902371411782946761303134397195657180991572207443365373111673136127
And then using high powered tools (PHP if you can believe it) I verified no more primes up to a(51201) But a(1024) is composite with smallest factor of more than 2 billion.
1024 is a power of two... 2^10 to be specific...
Perhaps I moved too fast. a() is the function for which this thread is created.
a(1024) = Ceiling( (½ + ½√5)^1024 )
= ½ + (½ + ½√5)^1024 + (½ − ½√5)^1024 − ½ (−1)^1024
= ½ + (½ + ½√5)^1024 + (−2/(1 + √5))^1024 − ½ (−1)^1024
= 10077286735077005660982008061065073068074475300466012444629388487574769652115651763500026128367679301744790365920278775601766000217455997930809875108639504578766853603625505162682177708433023235042368022152858871807
And it is not prime.
a(1024) = Ceiling( (½ + ½√5)^1024 )
= ½ + (½ + ½√5)^1024 + (½ − ½√5)^1024 − ½ (−1)^1024
= ½ + (½ + ½√5)^1024 + (−2/(1 + √5))^1024 − ½ (−1)^1024
= 10077286735077005660982008061065073068074475300466012444629388487574769652115651763500026128367679301744790365920278775601766000217455997930809875108639504578766853603625505162682177708433023235042368022152858871807
And it is not prime.
From one who has to "adjust" one's framings to get answers with more than 9 digits, just what exactly, program are you using?
I might add, for me and for others, is there a poor man's version of whatever that program is?
- RF
I might add, for me and for others, is there a poor man's version of whatever that program is?
- RF
I'm using PHP. It's cheap (free) -- runs on Linux and Windows -- and (important for our purposes) has arbitrary-precision math libraries.
<!DOCTYPE html
PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en-US" xml:lang="en-US">
<head>
<title>PHP Does Math!</title>
<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1" />
</head>
<body>
<h1>PHP and the Ceiling of powers of phi!</h1>
<?php
# A short list of primes which commonly divide values from our sequence
$m = array(2, 3, 5, 7, 11, 29, 47, 59, 127, 199, 223, 419, 449, 521,
607, 641, 1049, 1087, 1409, 2207, 3571, 4021, 6709, 21503, 34303, 58367,
89087, 119809, 1597441);
# Set up arrays to hold our recursion (modulo the above small primes)
$v = array(); $w = array(); $x = array(); $y = array();
foreach ( $m as $k ) {
$v[] = 2 % $k; $w[] = 3 % $k; $x[] = 5 % $k; $y[] = 7 % $k;
}
# Set up special variables to hold as many digits as we have memory
$vb = gmp_init(2); $wb = gmp_init(3); $xb = gmp_init(5); $yb = gmp_init(7);
# Set up watch for factors which repeat periodically.
$pat = array();
foreach ( $m as $k ) {
$pat[] = array(2 % $k, 3 % $k, 5 % $k, 7 % $k);
}
$period = 0;
# Compute a(5) ... a(10000)
for($i = 5; $i <= 10000; $i ++ ) {
# The following computes yb + 2 * xb - (wb + vb) for as many digits as
# we have memory
$b=gmp_sub(gmp_add($yb,gmp_mul($xb,2)),gmp_add($wb,$vb));
# We compute if this new element of the sequence is zero modulo one of
# our small primes.
$poss_prime = TRUE;
$z = array();
foreach( $m as $j => $k ) {
$z[$j] = ($y[$j] + 2* $x[$j] + 2 * $k - $w[$j] - $v[$j]) % $k;
# if any z (which is modulo k for each of our small primes) is zero,
# then the element of the sequence can only be a prime if is it equal to
# k -- but we already found those cases for small primes a(1) = 2, a(2)
# = 3, a(3) = 5, a(4)=7, a(8)=47, a(16)=2207. We are not interested in
# those small primes.
if ( $z[$j] == 0 ) {
$poss_prime = FALSE;
}
}
# Now that the z's are done -- shift all of the elements by one
$v = $w; $w = $x; $x = $y; $y = $z;
# Pattern detection -- if (modulo k -- a small prime) the saved sequence
# matches the original sequence then we have detected periodicity.
$period ++;
foreach( $m as $j => $k ) {
if (
$v[$j] == $pat[$j][0]
&& $w[$j] == $pat[$j][1]
&& $x[$j] == $pat[$j][2]
&& $z[$j] == $pat[$j][3] ) {
# Break the pattern so we can never match again
$pat[$j][0] = -1;
# Tell the user about it.
echo '<br />', 'Factor ', $k, ' has period ', $period, "\n";
}
}
# Remember $b -- it holds a large number -- we want to see if it is prime or not
if ( $poss_prime || $i == 8 || $i == 16 ) {
$pp = gmp_prob_prime($b);
if ( $pp == 2 ) {
echo '<br /> a(', $i, ') = ', gmp_strval($b);
echo '<br /> a(', $i, ') SURELY PRIME ', "\n";
} elseif ( $pp ) {
# It is very hard to prove a number is prime -- but if the computer is
# pretty sure we go here
echo '<br /> a(', $i, ') = ', gmp_strval($b);
echo '<br /> a(', $i, ') PROB PRIME ', "\n";
} else {
# It is often easy to prove a number is NOT prime
echo '<br /> a(', $i, ') = ', gmp_strval($b);
echo '<br /> a(', $i, ') NOT PRIME ', "\n";
# But humans would like to know if it has small factors
for ($k = 12; $k < 2000; $k += 6 ) {
$f = $k - 1;
if ( gmp_intval(gmp_mod($b, $f)) == 0 ) {
echo '<br /> a(', $i, ') FACTOR ', $f, "\n";
}
$f = $k + 1;
if ( gmp_intval(gmp_mod($b, $f)) == 0 ) {
echo '<br /> a(', $i, ') FACTOR ', $f, "\n";
}
}
echo "\n";
}
}
# Shift all the big numbers by one place -- just like we already did with the modulo numbers
$vb = $wb; $wb = $xb; $xb = $yb; $yb = $b;
}
?>
</body>
</html>
CODE
<!DOCTYPE html
PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" lang="en-US" xml:lang="en-US">
<head>
<title>PHP Does Math!</title>
<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1" />
</head>
<body>
<h1>PHP and the Ceiling of powers of phi!</h1>
<?php
# A short list of primes which commonly divide values from our sequence
$m = array(2, 3, 5, 7, 11, 29, 47, 59, 127, 199, 223, 419, 449, 521,
607, 641, 1049, 1087, 1409, 2207, 3571, 4021, 6709, 21503, 34303, 58367,
89087, 119809, 1597441);
# Set up arrays to hold our recursion (modulo the above small primes)
$v = array(); $w = array(); $x = array(); $y = array();
foreach ( $m as $k ) {
$v[] = 2 % $k; $w[] = 3 % $k; $x[] = 5 % $k; $y[] = 7 % $k;
}
# Set up special variables to hold as many digits as we have memory
$vb = gmp_init(2); $wb = gmp_init(3); $xb = gmp_init(5); $yb = gmp_init(7);
# Set up watch for factors which repeat periodically.
$pat = array();
foreach ( $m as $k ) {
$pat[] = array(2 % $k, 3 % $k, 5 % $k, 7 % $k);
}
$period = 0;
# Compute a(5) ... a(10000)
for($i = 5; $i <= 10000; $i ++ ) {
# The following computes yb + 2 * xb - (wb + vb) for as many digits as
# we have memory
$b=gmp_sub(gmp_add($yb,gmp_mul($xb,2)),gmp_add($wb,$vb));
# We compute if this new element of the sequence is zero modulo one of
# our small primes.
$poss_prime = TRUE;
$z = array();
foreach( $m as $j => $k ) {
$z[$j] = ($y[$j] + 2* $x[$j] + 2 * $k - $w[$j] - $v[$j]) % $k;
# if any z (which is modulo k for each of our small primes) is zero,
# then the element of the sequence can only be a prime if is it equal to
# k -- but we already found those cases for small primes a(1) = 2, a(2)
# = 3, a(3) = 5, a(4)=7, a(8)=47, a(16)=2207. We are not interested in
# those small primes.
if ( $z[$j] == 0 ) {
$poss_prime = FALSE;
}
}
# Now that the z's are done -- shift all of the elements by one
$v = $w; $w = $x; $x = $y; $y = $z;
# Pattern detection -- if (modulo k -- a small prime) the saved sequence
# matches the original sequence then we have detected periodicity.
$period ++;
foreach( $m as $j => $k ) {
if (
$v[$j] == $pat[$j][0]
&& $w[$j] == $pat[$j][1]
&& $x[$j] == $pat[$j][2]
&& $z[$j] == $pat[$j][3] ) {
# Break the pattern so we can never match again
$pat[$j][0] = -1;
# Tell the user about it.
echo '<br />', 'Factor ', $k, ' has period ', $period, "\n";
}
}
# Remember $b -- it holds a large number -- we want to see if it is prime or not
if ( $poss_prime || $i == 8 || $i == 16 ) {
$pp = gmp_prob_prime($b);
if ( $pp == 2 ) {
echo '<br /> a(', $i, ') = ', gmp_strval($b);
echo '<br /> a(', $i, ') SURELY PRIME ', "\n";
} elseif ( $pp ) {
# It is very hard to prove a number is prime -- but if the computer is
# pretty sure we go here
echo '<br /> a(', $i, ') = ', gmp_strval($b);
echo '<br /> a(', $i, ') PROB PRIME ', "\n";
} else {
# It is often easy to prove a number is NOT prime
echo '<br /> a(', $i, ') = ', gmp_strval($b);
echo '<br /> a(', $i, ') NOT PRIME ', "\n";
# But humans would like to know if it has small factors
for ($k = 12; $k < 2000; $k += 6 ) {
$f = $k - 1;
if ( gmp_intval(gmp_mod($b, $f)) == 0 ) {
echo '<br /> a(', $i, ') FACTOR ', $f, "\n";
}
$f = $k + 1;
if ( gmp_intval(gmp_mod($b, $f)) == 0 ) {
echo '<br /> a(', $i, ') FACTOR ', $f, "\n";
}
}
echo "\n";
}
}
# Shift all the big numbers by one place -- just like we already did with the modulo numbers
$vb = $wb; $wb = $xb; $xb = $yb; $yb = $b;
}
?>
</body>
</html>
The output looks something like this:
QUOTE
PHP and the Ceiling of powers of phi!
Factor 5 has period 4
a(8) = 47
a(8) SURELY PRIME
Factor 2 has period 6
Factor 3 has period 8
Factor 11 has period 10
a(16) = 2207
a(16) SURELY PRIME
Factor 29 has period 14
Factor 7 has period 16
Factor 199 has period 22
Factor 521 has period 26
Factor 47 has period 32
Factor 3571 has period 34
Factor 59 has period 58
Factor 2207 has period 64
Factor 6709 has period 86
Factor 1087 has period 128
Factor 4021 has period 134
Factor 127 has period 256
Factor 1049 has period 262
Factor 419 has period 418
Factor 223 has period 448
Factor 449 has period 448
Factor 119809 has period 512
Factor 641 has period 640
Factor 1409 has period 704
a(1024) = 10077286735077005660982008061065073068074475300466012444629388487574769652115651763500026128367679301744790365920278775601766000217455997930809875108639504578766853603625505162682177708433023235042368022152858871807
a(1024) NOT PRIME
Factor 34303 has period 1024
Factor 607 has period 1216
a(2048) = 101551707940958976476873723686691537560124936216049957128359991876726343954548826671188589246207404049249584475544225686479598181242824239865026718975582831790339042655215629893459077413144429144815676624628680986839974189561981371422914234508007983748918604313893297542075331365600286467195288043916251150089236097377941252734677327147814393642121714369673795261602383375518218563568418129690432349780011400594146198108859445247
a(2048) NOT PRIME
Factor 21503 has period 2048
a(4096) = 10312749385725830441802540740065515313851581175704414324490412533525451490068004056562555361964908876484176036910001863956624258046666641669263753146501441696735313929679410169743156859880792433984484789744611483427989571676851808465471269623465120455949137399300388899881375908805357877664472786728973959341025009696822210060021549414146476712439210747763775120130672397433546451190984800181655778096308930673967929247719493575654387577417021188689832435468683226810568309660909493278434794372904636841074705443843966438365619077399335713902545305125588892910321945577013801232719970629342173590182755151700994830811432603344551390194800209686663531108157905060696146769539632464420378593090076812946660899750748092666861150545474346290900031927298783752074470630435760298639123733285580454340205109167983980835350647593511713883686631638820631484590891007
a(4096) NOT PRIME
a(5120) = 103924532586948449541025548469905042815163160870538791878076860982467729213967638999936756132786738637140938498926051582961868195686705886156988403152278449048401999206924810190200305244985190654713851693117398830457531422075694620069014519833369298344191759398465110058999822873175751169891968814421429001793800697375899372768454830574609743352221048964027011884102587659972293183495442940142668694477320385247942177709810098996340772082694812799631146215964847590062461621745003714104865232344469044025174047318740665856173794198248104589632844731622776699231900112053393308566101393320861092316797352649667932816354201965541042937052661738709828843406764314500987421590979349766403019466801795506751071373538223916947264969266681899936518248324237027149334888507663175560653336390261986392890649689013762172320344653833798734635974223635552271785351325998675644470204905152019184474983185413232192079513550786691928303381357479705268196707975564212783799081591688869304828599617169613510402983341179650289869627170768655378114874743115898574776896525150074439472560127
a(5120) NOT PRIME
a(7168) = 10553713781170462785747033859778190717923554053112560653871468909699552194674189684847348431150849756135559079413277813782980675185576204841605353564418352865161132618666782942596769810765713567436786301098324940516327910006397645217127488139893665666699415292685691906293854378920700227414924931953903972193284769541515553999583309850450085223535769272383403183963283525178049100375341336985167604043630237555517809508224538932811585369919519258357342577818846414487480059353615738759373269534877646332445519742988030599344592962806743218283652205976659247969799182254601680362883649409581229415637057382602650383309340159315915107611080867280322725207485041832944987106446768151867792747854931506803663085150308482911851730709183293325578710034493683446505969816444799836464907928470776290538863787235297782662461762131696124199496534850016758682242861474871823560295761546414537054808743643605332457519657687504566645892258476817279523465351032245977793685696711382646936628433180525585204625289756116346386727596918815977647441761738938189418741115650028026235469750740597455605273121065530870253887884002311139470898951851028693841872653063050427927276394216167734241108324798187918356731304485584841310997005928491542387373026891328895166607250890085613618769256250708667213409098081818687187305953503537216272882187157427334717817608399966327815061763609633646028469764455653038756140052464317147509276429496760490466626555176845032950051327913049628483729892383555286099256289336864622286847
a(7168) NOT PRIME
a(8192) = 106352799892788493110153690447343447305470461839801749696888904326793822501807523858343368817374906071055883020948513173454934275225964193652567805803093812427396216375497396011426701253537040886005058364455749508379332460068863273209362959211349695940992965287501020465878192236379910996201246410952503288582362187697323696152715258347507554225929317934354812864403928775594189364372534345190756371310821050130456496752730896560557090600635970693008450571523015564151725610119684508144165403330461173232751131580337443954064658145831159957490227960266786882241002797161000132349989681771373659042440825339587753496034215354174180882342606891519666762174814589937832916347605938886393346759579533095469236801706417967993461442257672990927199494961969540216293178443558375799261101214910353496376696966116034407448019538663751360223550121654027258862014313329045408270050541733443965519859471647956029416780580658494748034461836394695473721726251355786224135077600672018223272986827418873344211259914685891948265104665410891785898850985208482867129812960455716871960916151438290908981447348308505752069195875345801164338971873752388784257588390965689486787558399481835958463278218442281288972079544765108396832760769088884560324747007796054306029913968807790745720370504594274537869265965934894432425875393151213643849488026856220837544470554233075366408557809059873632701041552713034095222195724966458636832609260553157412414706490526236978187501358385792586601605835876189691641049462810938225466524678771277916462642973015490018279647696101595540712249410013175032489879191921472731396498012877730568542083001326763588214716437813175362270505999306955046622201907495073417434834966899533608716262480958153474047
a(8192) NOT PRIME
Factor 5 has period 4
a(8) = 47
a(8) SURELY PRIME
Factor 2 has period 6
Factor 3 has period 8
Factor 11 has period 10
a(16) = 2207
a(16) SURELY PRIME
Factor 29 has period 14
Factor 7 has period 16
Factor 199 has period 22
Factor 521 has period 26
Factor 47 has period 32
Factor 3571 has period 34
Factor 59 has period 58
Factor 2207 has period 64
Factor 6709 has period 86
Factor 1087 has period 128
Factor 4021 has period 134
Factor 127 has period 256
Factor 1049 has period 262
Factor 419 has period 418
Factor 223 has period 448
Factor 449 has period 448
Factor 119809 has period 512
Factor 641 has period 640
Factor 1409 has period 704
a(1024) = 10077286735077005660982008061065073068074475300466012444629388487574769652115651763500026128367679301744790365920278775601766000217455997930809875108639504578766853603625505162682177708433023235042368022152858871807
a(1024) NOT PRIME
Factor 34303 has period 1024
Factor 607 has period 1216
a(2048) = 101551707940958976476873723686691537560124936216049957128359991876726343954548826671188589246207404049249584475544225686479598181242824239865026718975582831790339042655215629893459077413144429144815676624628680986839974189561981371422914234508007983748918604313893297542075331365600286467195288043916251150089236097377941252734677327147814393642121714369673795261602383375518218563568418129690432349780011400594146198108859445247
a(2048) NOT PRIME
Factor 21503 has period 2048
a(4096) = 10312749385725830441802540740065515313851581175704414324490412533525451490068004056562555361964908876484176036910001863956624258046666641669263753146501441696735313929679410169743156859880792433984484789744611483427989571676851808465471269623465120455949137399300388899881375908805357877664472786728973959341025009696822210060021549414146476712439210747763775120130672397433546451190984800181655778096308930673967929247719493575654387577417021188689832435468683226810568309660909493278434794372904636841074705443843966438365619077399335713902545305125588892910321945577013801232719970629342173590182755151700994830811432603344551390194800209686663531108157905060696146769539632464420378593090076812946660899750748092666861150545474346290900031927298783752074470630435760298639123733285580454340205109167983980835350647593511713883686631638820631484590891007
a(4096) NOT PRIME
a(5120) = 103924532586948449541025548469905042815163160870538791878076860982467729213967638999936756132786738637140938498926051582961868195686705886156988403152278449048401999206924810190200305244985190654713851693117398830457531422075694620069014519833369298344191759398465110058999822873175751169891968814421429001793800697375899372768454830574609743352221048964027011884102587659972293183495442940142668694477320385247942177709810098996340772082694812799631146215964847590062461621745003714104865232344469044025174047318740665856173794198248104589632844731622776699231900112053393308566101393320861092316797352649667932816354201965541042937052661738709828843406764314500987421590979349766403019466801795506751071373538223916947264969266681899936518248324237027149334888507663175560653336390261986392890649689013762172320344653833798734635974223635552271785351325998675644470204905152019184474983185413232192079513550786691928303381357479705268196707975564212783799081591688869304828599617169613510402983341179650289869627170768655378114874743115898574776896525150074439472560127
a(5120) NOT PRIME
a(7168) = 10553713781170462785747033859778190717923554053112560653871468909699552194674189684847348431150849756135559079413277813782980675185576204841605353564418352865161132618666782942596769810765713567436786301098324940516327910006397645217127488139893665666699415292685691906293854378920700227414924931953903972193284769541515553999583309850450085223535769272383403183963283525178049100375341336985167604043630237555517809508224538932811585369919519258357342577818846414487480059353615738759373269534877646332445519742988030599344592962806743218283652205976659247969799182254601680362883649409581229415637057382602650383309340159315915107611080867280322725207485041832944987106446768151867792747854931506803663085150308482911851730709183293325578710034493683446505969816444799836464907928470776290538863787235297782662461762131696124199496534850016758682242861474871823560295761546414537054808743643605332457519657687504566645892258476817279523465351032245977793685696711382646936628433180525585204625289756116346386727596918815977647441761738938189418741115650028026235469750740597455605273121065530870253887884002311139470898951851028693841872653063050427927276394216167734241108324798187918356731304485584841310997005928491542387373026891328895166607250890085613618769256250708667213409098081818687187305953503537216272882187157427334717817608399966327815061763609633646028469764455653038756140052464317147509276429496760490466626555176845032950051327913049628483729892383555286099256289336864622286847
a(7168) NOT PRIME
a(8192) = 106352799892788493110153690447343447305470461839801749696888904326793822501807523858343368817374906071055883020948513173454934275225964193652567805803093812427396216375497396011426701253537040886005058364455749508379332460068863273209362959211349695940992965287501020465878192236379910996201246410952503288582362187697323696152715258347507554225929317934354812864403928775594189364372534345190756371310821050130456496752730896560557090600635970693008450571523015564151725610119684508144165403330461173232751131580337443954064658145831159957490227960266786882241002797161000132349989681771373659042440825339587753496034215354174180882342606891519666762174814589937832916347605938886393346759579533095469236801706417967993461442257672990927199494961969540216293178443558375799261101214910353496376696966116034407448019538663751360223550121654027258862014313329045408270050541733443965519859471647956029416780580658494748034461836394695473721726251355786224135077600672018223272986827418873344211259914685891948265104665410891785898850985208482867129812960455716871960916151438290908981447348308505752069195875345801164338971873752388784257588390965689486787558399481835958463278218442281288972079544765108396832760769088884560324747007796054306029913968807790745720370504594274537869265965934894432425875393151213643849488026856220837544470554233075366408557809059873632701041552713034095222195724966458636832609260553157412414706490526236978187501358385792586601605835876189691641049462810938225466524678771277916462642973015490018279647696101595540712249410013175032489879191921472731396498012877730568542083001326763588214716437813175362270505999306955046622201907495073417434834966899533608716262480958153474047
a(8192) NOT PRIME
QUOTE (Raphie Frank+Mar 18 2009, 02:47 AM)
And anyone who judges, much less censors, those exploring new methods and means to acquire knowledge is also far less than wise. - RF
Bukh said it best IMHO.
Bukh said it best IMHO.
QUOTE (bukh+Feb 16 2009, 12:12 AM)
I stumbled over these Posting Guidelines put forward in another Discussion Forum. And I am not going to revealing which one.
"Sciences
There will be no discussions of highly speculative subjects in the Science Forums.
Topics or responses dealing with the pixel universe, the æther, or other perennially resurgent pseudo-science will be deleted.
If a member abuses these guidelines their posting permissions for this forum will be removed, either for a probationary period or permanently, at the administrator's discretion."
Let us pray to that Mother Nature follow Scientific rules and regulations.
The unmentioned forum's rules were misquoted somehow.
There has been a revision, however:
So ⌈φⁿ⌉ = 0 mod 47 when n = 8 or 17 or 24 or 31 mod 32
Conjecture II: Corollary to Conjecture I holds for all initial conditions for a recurrence of a(n+4) = a(n+3) + 2 a(n+2) - a(n+1) - a(n)
Conjecture III: Corollary to Conjecture I holds for all initial conditions for all linear recurrences where the integer coefficients sum to one.
Conjecture IV: Conjecture III (which implies all of the above) is in the literature.
//Edit -- Conjectures II through IV are Dead to me. Conjecture I looks harder to prove than I hoped.
Demonstration of failure of Conjecture II
"Sciences
There will be no discussions of highly speculative subjects in the Science Forums.
Topics or responses dealing with the pixel universe, the æther, or other perennially resurgent pseudo-science will be deleted.
If a member abuses these guidelines their posting permissions for this forum will be removed, either for a probationary period or permanently, at the administrator's discretion."
Let us pray to that Mother Nature follow Scientific rules and regulations.
The unmentioned forum's rules were misquoted somehow.
There has been a revision, however:QUOTE
Sciences
There will be no discussions of highly speculative subjects in the Science Forums.
Topics or responses dealing with the pixel universe, the æther, or other perennially resurgent pseudo-science will be deleted.
Science Forum Posting Rules - please read!
When posting in any of the science forums (with the exception of the speculations forum), please remember to keep the content to mainstream science only. Any questionable or debatable material needs to be backed up with a link to a reputable study or article.
When speculating in a scientific thread, please let it be known that it is speculation, and keep it to an absolute minimum to keep the thread from going off track.
If a member abuses these guidelines their posting permissions for this forum will be removed, either for a probationary period or permanently, at the administrator's discretion.
There will be no discussions of highly speculative subjects in the Science Forums.
Topics or responses dealing with the pixel universe, the æther, or other perennially resurgent pseudo-science will be deleted.
Science Forum Posting Rules - please read!
When posting in any of the science forums (with the exception of the speculations forum), please remember to keep the content to mainstream science only. Any questionable or debatable material needs to be backed up with a link to a reputable study or article.
When speculating in a scientific thread, please let it be known that it is speculation, and keep it to an absolute minimum to keep the thread from going off track.
If a member abuses these guidelines their posting permissions for this forum will be removed, either for a probationary period or permanently, at the administrator's discretion.
QUOTE (rpenner+Mar 20 2009, 12:21 PM)
Perhaps I moved too fast. a() is the function for which this thread is created.
a(1024) = Ceiling( (½ + ½?5)^1024 )
= ½ + (½ + ½?5)^1024 + (½ ? ½?5)^1024 ? ½ (?1)^1024
= ½ + (½ + ½?5)^1024 + (?2/(1 + ?5))^1024 ? ½ (?1)^1024
= 10077286735077005660982008061065073068074475300466012444629388487574769652115651763500026128367679301744790365920278775601766000217455997930809875108639504578766853603625505162682177708433023235042368022152858871807
And it is not prime.
I moved too fast in my reply. My bad.
Fascinating posts RPenner. Thank you.
If I understand correctly, the following line:
a(7)>3000 or non-existent. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 07 2007
could now be updated to read:
a(7)>51201 or non-existent. - RPenner March 20, 2009
Best,
Raphie
a(1024) = Ceiling( (½ + ½?5)^1024 )
= ½ + (½ + ½?5)^1024 + (½ ? ½?5)^1024 ? ½ (?1)^1024
= ½ + (½ + ½?5)^1024 + (?2/(1 + ?5))^1024 ? ½ (?1)^1024
= 10077286735077005660982008061065073068074475300466012444629388487574769652115651763500026128367679301744790365920278775601766000217455997930809875108639504578766853603625505162682177708433023235042368022152858871807
And it is not prime.
I moved too fast in my reply. My bad.
Fascinating posts RPenner. Thank you.
If I understand correctly, the following line:
a(7)>3000 or non-existent. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 07 2007
could now be updated to read:
a(7)>51201 or non-existent. - RPenner March 20, 2009
Best,
Raphie
Granouille
"QUOTE (Raphie Frank @ Mar 18 2009, 02:47 AM)
And anyone who judges, much less censors, those exploring new methods and means to acquire knowledge is also far less than wise. - RF
Bukh said it best IMHO.
QUOTE (bukh @ Feb 16 2009, 12:12 AM)
I stumbled over these Posting Guidelines put forward in another Discussion Forum. And I am not going to revealing which one.
"Sciences
There will be no discussions of highly speculative subjects in the Science Forums.
Topics or responses dealing with the pixel universe, the æther, or other perennially resurgent pseudo-science will be deleted.
If a member abuses these guidelines their posting permissions for this forum will be removed, either for a probationary period or permanently, at the administrator's discretion."
Let us pray to that Mother Nature follow Scientific rules and regulations.
The unmentioned forum's rules were misquoted somehow. There has been a revision, however:"
Granouille - now that you have undertaken the role as defender for said Forums rather bizare rules - let me ask you why you are saying that they "were misquoted somehow." ??
Are you indicating that the rules were differently phrased - and that they have been quoted in an uncomplete or changed form - then my answer is NO - they have been EXACTLY quoted.
Are you talking out from ignorance or are you trying to paint the picture less stupid and bizare ? and in the latter case - Why ?
It is a serious act of dishonesty trying to bend the truth.
"QUOTE (Raphie Frank @ Mar 18 2009, 02:47 AM)
And anyone who judges, much less censors, those exploring new methods and means to acquire knowledge is also far less than wise. - RF
Bukh said it best IMHO.
QUOTE (bukh @ Feb 16 2009, 12:12 AM)
I stumbled over these Posting Guidelines put forward in another Discussion Forum. And I am not going to revealing which one.
"Sciences
There will be no discussions of highly speculative subjects in the Science Forums.
Topics or responses dealing with the pixel universe, the æther, or other perennially resurgent pseudo-science will be deleted.
If a member abuses these guidelines their posting permissions for this forum will be removed, either for a probationary period or permanently, at the administrator's discretion."
Let us pray to that Mother Nature follow Scientific rules and regulations.
The unmentioned forum's rules were misquoted somehow. There has been a revision, however:"
Granouille - now that you have undertaken the role as defender for said Forums rather bizare rules - let me ask you why you are saying that they "were misquoted somehow." ??
Are you indicating that the rules were differently phrased - and that they have been quoted in an uncomplete or changed form - then my answer is NO - they have been EXACTLY quoted.
Are you talking out from ignorance or are you trying to paint the picture less stupid and bizare ? and in the latter case - Why ?
It is a serious act of dishonesty trying to bend the truth.
What I still like about that forum despite having been temporarily banned few times is that it is civilized. And moderators work real hard to keep it like that.
So , few notorious people from here do not appear there exactly because they do not pass "civilized" test.
The price to be paid is to be more accurate and show progress in understanding- in some sense of course to please the moderators- but that is their right, and they are learning as well- LEARNING and changing.
Monkeys typing on typing machines will never produce Hamlet.
Monkeys with feedback ( each time they deviate from Hamlet too much they get hit with electric current ) may produce something usable:)
I know -bukh -personally You may disagree from Your experience. However, for the readability of forum and for positioning of forum so that more certain level people get atracted to it, quality of individual posts and threads, even speculative ones , has to be very carefully maintained, and I appreciate it. I suggest You show more focus on results than on priciples. I am reading physics/math like hell (difficult!) to get some improvement in discussions, and I moderate down my visionary "general absolute truth" statements which anyway change after few months . People do not like to discuss such statements, and rightly so.
See what rpenner has done to this forum. It is undoubtelly a plus, if he would just add to his moderation techniques also requirement for higher civilization standards...
I would suggest an automatic limit to post length to improve dialogue. Perhaps 2000 marks or so, plus add LaTex.
For example, this post I just made was too long.
So , few notorious people from here do not appear there exactly because they do not pass "civilized" test.
The price to be paid is to be more accurate and show progress in understanding- in some sense of course to please the moderators- but that is their right, and they are learning as well- LEARNING and changing.
Monkeys typing on typing machines will never produce Hamlet.
Monkeys with feedback ( each time they deviate from Hamlet too much they get hit with electric current ) may produce something usable:)
I know -bukh -personally You may disagree from Your experience. However, for the readability of forum and for positioning of forum so that more certain level people get atracted to it, quality of individual posts and threads, even speculative ones , has to be very carefully maintained, and I appreciate it. I suggest You show more focus on results than on priciples. I am reading physics/math like hell (difficult!) to get some improvement in discussions, and I moderate down my visionary "general absolute truth" statements which anyway change after few months . People do not like to discuss such statements, and rightly so.
See what rpenner has done to this forum. It is undoubtelly a plus, if he would just add to his moderation techniques also requirement for higher civilization standards...
I would suggest an automatic limit to post length to improve dialogue. Perhaps 2000 marks or so, plus add LaTex.
For example, this post I just made was too long.
QUOTE (rpenner+Mar 20 2009, 01:07 AM)
Modulo 2 it is periodic of period 6
Modulo 3 it is periodic of period 8
Modulo 5 it is periodic of period 4
Modulo 7 it is periodic of period 16
Modulo ***47 it is periodic of period 32***
RP, this threw me off completely and I have been itching to ask: shouldn't it be 10?
(It doesn't make sense to me! It "just should" be 10!)
Modulo 3 it is periodic of period 8
Modulo 5 it is periodic of period 4
Modulo 7 it is periodic of period 16
Modulo ***47 it is periodic of period 32***
RP, this threw me off completely and I have been itching to ask: shouldn't it be 10?
(It doesn't make sense to me! It "just should" be 10!)
You don't state why it should be 10?
Modulo 47 it is 32.
Modulo 47 it is 32.
CODE
1 2 9 30 17 0 25 19 33 2
2 3 10 29 18 44 26 18 34 3
3 5 11 12 19 44 27 37 34 5
4 7 12 40 20 40 28 7 35 7
5 12 13 5 21 37 29 44 .. ..
6 18 14 44 22 29 30 3 .. ..
7 30 15 2 23 19 31 0 .. ..
8 0 16 45 24 0 32 2 .. ..
2 3 10 29 18 44 26 18 34 3
3 5 11 12 19 44 27 37 34 5
4 7 12 40 20 40 28 7 35 7
5 12 13 5 21 37 29 44 .. ..
6 18 14 44 22 29 30 3 .. ..
7 30 15 2 23 19 31 0 .. ..
8 0 16 45 24 0 32 2 .. ..
So ⌈φⁿ⌉ = 0 mod 47 when n = 8 or 17 or 24 or 31 mod 32
Conjecture I: If for prime p, k is the smallest natural number such that a(n) ≡ a(n+k) mod p for all n, then 2p² ≡ 2 mod k.
Corollary: a(n) modulo p is periodic with period 2p² - 2 (though this may be far from the shortest description of the periodicity).
Examples:
Corollary: a(n) modulo p is periodic with period 2p² - 2 (though this may be far from the shortest description of the periodicity).
Examples:
CODE
p k (2p²-2)/k
5 4 12
2 6 1
11 10 24
7 16 6
19 18 40
17 36 16
5 4 12
2 6 1
11 10 24
7 16 6
19 18 40
17 36 16
Conjecture II: Corollary to Conjecture I holds for all initial conditions for a recurrence of a(n+4) = a(n+3) + 2 a(n+2) - a(n+1) - a(n)
Conjecture III: Corollary to Conjecture I holds for all initial conditions for all linear recurrences where the integer coefficients sum to one.
Conjecture IV: Conjecture III (which implies all of the above) is in the literature.
//Edit -- Conjectures II through IV are Dead to me. Conjecture I looks harder to prove than I hoped.
Demonstration of failure of Conjecture II
CODE
( 1 2 -1 -1 )^20 = ( 1 0 0 0 ) mod 5
. 1 0 0 0 0 1 0 0
. 0 1 0 0 0 0 1 0
. 0 0 1 0 0 0 0 1
. 1 0 0 0 0 1 0 0
. 0 1 0 0 0 0 1 0
. 0 0 1 0 0 0 0 1
CODE
( 1 2 -1 -1 )^4 = ( -2 -1 -2 1 ) mod 5
. 1 0 0 0 -1 -1 1 2
. 0 1 0 0 -2 1 -2 -1
. 0 0 1 0 1 2 -1 -1
. 1 0 0 0 -1 -1 1 2
. 0 1 0 0 -2 1 -2 -1
. 0 0 1 0 1 2 -1 -1
Observation: a(n) = 3u(n-1) + u(n-2) + (1 - (-1)^n)/2 where u() is the Fibonacci sequence with u(0)=0.
So we can leverage off the literature:
D.W. Robinson, "The Fibonacci matrix modulo m" Fibonacci Quarterly. 1 29-36 (1963)
J.D. Fulton, W.L. Morris "On arithmetical functions related to the Fibonacci numbers" Acta. Arithmetica XVI 105-110 (1969)
Where π(m) is the periodicity of the Fibonacci sequence taken modulo m.
We have the following result:
π(5) = 20
π(p)|(p-1) if p is a prime number = ±1 mod 5
π(p)|(2p+2) if p is a prime number = ±2 mod 5
So the simple fact that the periodicity of (-1)^n is 2, gives us k(p)|LCM(2,π(p))
or
k(5) = 4 and 4|(5-1) and 4|(2×5+2)
k(p)|(p-1) if p is a prime number = ±1 mod 5
k(p)|(2p+2) if p is a prime number = ±2 mod 5
which gives us:
If q is a prime and m > 2 and q|k(m), then q<m.
And even more trivially: k(p)|(2p² - 2)
And because a(-1) (working with the recursion, not the original definition) is zero, then hopes for prime values above a(16) grow very slim. Best hopes are with numbers of the form a(2^n) for which there is an easy recursion:
a(2) = 3, a(2^(n+1)) = (a(2^n))^2 - 2
a(8192) is composite but has resisted 3 days of ECM factorization on my pityful hardware.
So we can leverage off the literature:
D.W. Robinson, "The Fibonacci matrix modulo m" Fibonacci Quarterly. 1 29-36 (1963)
J.D. Fulton, W.L. Morris "On arithmetical functions related to the Fibonacci numbers" Acta. Arithmetica XVI 105-110 (1969)
Where π(m) is the periodicity of the Fibonacci sequence taken modulo m.
We have the following result:
π(5) = 20
π(p)|(p-1) if p is a prime number = ±1 mod 5
π(p)|(2p+2) if p is a prime number = ±2 mod 5
So the simple fact that the periodicity of (-1)^n is 2, gives us k(p)|LCM(2,π(p))
or
k(5) = 4 and 4|(5-1) and 4|(2×5+2)
k(p)|(p-1) if p is a prime number = ±1 mod 5
k(p)|(2p+2) if p is a prime number = ±2 mod 5
which gives us:
If q is a prime and m > 2 and q|k(m), then q<m.
And even more trivially: k(p)|(2p² - 2)
And because a(-1) (working with the recursion, not the original definition) is zero, then hopes for prime values above a(16) grow very slim. Best hopes are with numbers of the form a(2^n) for which there is an easy recursion:
a(2) = 3, a(2^(n+1)) = (a(2^n))^2 - 2
a(8192) is composite but has resisted 3 days of ECM factorization on my pityful hardware.
QUOTE (Raphie Frank+Mar 18 2009, 08:47 AM)
P.S. There is a moral to this story, and I will make it very simple. NOONE will be able to answer this question. Of this I am quite certain. Point being, in no way, shape or form do we have it all figured out, and anyone who thinks otherwise is a fool.
Dear RPenner,
I would be very happy for you to prove me wrong.
Best,
Raphie
Dear RPenner,
I would be very happy for you to prove me wrong.
Best,
Raphie
Dear RPenner,
A related but different question. For the following:
{Ceiling 3*TETRA_k/2 | Ceiling (3*(TETRA_k)/2 = 2^n -2}
I see the following solutions...
x = - 1, 0, 1, 2, 4, 7
where...
TETRA_x = 0, 0, 1, 4, 20, 84
Ceiling 3*(TETRA_x)/2 = 0, 0, 2, 6, 30, 126
One solution for x (amongst many), on the basis of known elements for this set of nunbers, where F_k = kth Fibonacci Number (START F_0) is x = sigma_(F_(k-2)); RANGE (F_0 --> F_6)
I am wondering if you would be willing to bring your number theoretic skills to bear in terms of ascertaining whether or not there are any additional values for x.
As opposed to the initial question upon this thread, I believe this one to be easily provable for someone with proper techniques at his command as have you...
My interest in this specific series relates to the fact that:
Ceiling 3*(TETRA_k)/2 = 0, 0, 0, 2, 6, 30, 126
... gives the complete set of (known) Double Mersenne Numbers (including Unity tripled...) , less one. That each of these integers is equal to a Kissing Number, or the Harmonic Mean of two Kissing Numbers ("Harmonic Kissing Number" = HK_n), I would hope, will not prejudice you against answering the question...
As an FYI, I have already checked all values up to x = 2048 and change. It can furthermore be considered "obvious" that no x = to a power of 2 will return a value equal to 2^n -2.
Best,
Raphie
NOTE: 3*(TETRA_k)/2 = (n^3 - n)/4
A related but different question. For the following:
{Ceiling 3*TETRA_k/2 | Ceiling (3*(TETRA_k)/2 = 2^n -2}
I see the following solutions...
x = - 1, 0, 1, 2, 4, 7
where...
TETRA_x = 0, 0, 1, 4, 20, 84
Ceiling 3*(TETRA_x)/2 = 0, 0, 2, 6, 30, 126
One solution for x (amongst many), on the basis of known elements for this set of nunbers, where F_k = kth Fibonacci Number (START F_0) is x = sigma_(F_(k-2)); RANGE (F_0 --> F_6)
I am wondering if you would be willing to bring your number theoretic skills to bear in terms of ascertaining whether or not there are any additional values for x.
As opposed to the initial question upon this thread, I believe this one to be easily provable for someone with proper techniques at his command as have you...
My interest in this specific series relates to the fact that:
Ceiling 3*(TETRA_k)/2 = 0, 0, 0, 2, 6, 30, 126
... gives the complete set of (known) Double Mersenne Numbers (including Unity tripled...) , less one. That each of these integers is equal to a Kissing Number, or the Harmonic Mean of two Kissing Numbers ("Harmonic Kissing Number" = HK_n), I would hope, will not prejudice you against answering the question...
As an FYI, I have already checked all values up to x = 2048 and change. It can furthermore be considered "obvious" that no x = to a power of 2 will return a value equal to 2^n -2.
Best,
Raphie
NOTE: 3*(TETRA_k)/2 = (n^3 - n)/4
Somehow I just know that somewhere Euler is bashing his head into his keyboard in exasperation......
QUOTE (AlphaNumeric+Aug 22 2009, 11:43 AM)
Somehow I just know that somewhere Euler is bashing his head into his keyboard in exasperation......
First, AlphaNumeric (and RPenner), let me note that x = -2 is also a solution, so the question is if there are any more solutions to the equation, 1.5Y = 2^n - 2, than -2,-1, 0, 1, 2, 4, 7, where Y equals some Tetrahedral Number.
Second, if the question is "bash-worthy," there can only be two reasonable explanations why:
A ) The solution is blindingly obvious; or...
B ) Because of the manner in which I came to ask the question, via my explorations into Kissing Numbers, Harmonic or otherwise, a point I will eventually delve back into upon the "Numerology" thread in relation to RPenner's assertions of "nonsense," assertions I believe to be, themselves, utter nonsense.
In relation to B ), for now, let's simply stick to the question at hand and "store away" any extraneous comments related to process, Kissing Numbers, Fibonacci, etc...
Regarding, A ) I do not not believe the answer any more "obvious" (but could be wrong) than the fact that 4095 is the largest Triangular number equal to a 2^n - 1, not unrelated to the Ramanujan-Nagell Triangular Numbers: 0, 1, 3, 15, 4095.
So let me restate the question, because the question I asked is quite related:
Let...
Z_1 = Ceiling (2*X) = 2^n - 2
Z_2 = Ceiling (1.5*Y) = 2^n - 2
For...
======================================================
X = (equivalencies...)
------------------------
C ((k), (k-2)) ===== Binomial Notation formulation
T_(k-1) ===== n-Hedral formulation (T = Triangular Number)
(k^2 - k)/2! ===== Algebraic formulation
S [1,1;0,1]:[3,(k+1)] ===== Personal Notation (can be re-explained another time...)
Condition: k is greater than or equal to 1
Y = (equivalencies...)
------------------------
C ((k+1), (k-2)) ===== Binomial Notation formulation
TETRA_(k-1) ===== n-Hedral formulation (Tetra = Tetrahedral Number)
(k^3 - k)/3! ===== Algebraic formulation
S [1,1;0,1]:[4,(k+1)] ===== Personal Notation (can be re-explained another time...)
Condition: k is greater than or equal to 1
======================================================
QUESTIONS:
I. What are the solutions for k of X [k(x)] and k of Y [k(Y)]?
II. What are the values for n of X [n(x)] and n of Y [n(Y)]?
III. What are the values for X and Y?
IV. What are the values for Z_1 and Z_2
The possible solutions to Z_1, an eminently finite set, are already known:
k(X) = 1, 2, 3, 6, 91
n(X) = 1, 2, 3, 5, 13
X = 0, 1, 3, 15, 4095
Z_1 = 0, 2, 6, 30, 8190
The values for X are otherwise known as, yes, you possibly guessed it, the Ramanujan-Nagell Triangular Numbers.
The set of solutions for Z_2, to the best of my knowledge, is unknown, but here is what I have so far:
k(Y) = 1, 2, 3, 5, 8
n(Y) = 1, 2, 3, 5, 7
Y = 0, 1, 4, 20, 84
Z_1 = 0, 2, 6, 30, 126
Is this also a finite set?
I believe it may well be. If so, then...
{Z_1 U Z_2} = 0, 2, 6, 30, 126, 8190
In order, Unity + the first 5 Mersenne Primes, less 1.
But I could very well, of course, be mistaken. There may, in fact, be an infinite number of solutions for Z_2. Thus, my question.
I might add that, given that Nagell proved Z_1, confirming one of many Ramanujan Connjectures, it is reasonable to assume that any proofs for Z_2 would assume a similar character.
But, again, I could be wrong...
Best Regards,
Raphie
P.S. The form 2^n - 2, by the way, is related to a mathematical concept known as "Sphere Line Picking." http://mathworld.wolfram.com/SphereLinePicking.html
P.P.S. The Ramanujan-Nagell Squares are equal to 8*X + 1 = 8 * {0, 1, 3, 15, 4095} + 1 = 1, 9, 25, 121, 32761, a number set that, surprisingly, is not to be found on OEIS. Each Ramanujan-Nagell Square follows the form 2^n - 7. The indices for these square numbers are equal to (2 * k(X)) - 1) = 1, 3, 5, 11, 181.
RELATED SEQUENCES
A060728
http://www.research.att.com/~njas/sequences/A060728
Numbers n such that Ramanujan's equation x^2 + 7 = 2^n has an integer solution.
3, 4, 5, 7, 15
A038198
http://www.research.att.com/~njas/sequences/A038198
Numbers n such that n^2+7 is a power of 2.
1, 3, 5, 11, 181
First, AlphaNumeric (and RPenner), let me note that x = -2 is also a solution, so the question is if there are any more solutions to the equation, 1.5Y = 2^n - 2, than -2,-1, 0, 1, 2, 4, 7, where Y equals some Tetrahedral Number.
Second, if the question is "bash-worthy," there can only be two reasonable explanations why:
A ) The solution is blindingly obvious; or...
B ) Because of the manner in which I came to ask the question, via my explorations into Kissing Numbers, Harmonic or otherwise, a point I will eventually delve back into upon the "Numerology" thread in relation to RPenner's assertions of "nonsense," assertions I believe to be, themselves, utter nonsense.
In relation to B ), for now, let's simply stick to the question at hand and "store away" any extraneous comments related to process, Kissing Numbers, Fibonacci, etc...
Regarding, A ) I do not not believe the answer any more "obvious" (but could be wrong) than the fact that 4095 is the largest Triangular number equal to a 2^n - 1, not unrelated to the Ramanujan-Nagell Triangular Numbers: 0, 1, 3, 15, 4095.
So let me restate the question, because the question I asked is quite related:
Let...
Z_1 = Ceiling (2*X) = 2^n - 2
Z_2 = Ceiling (1.5*Y) = 2^n - 2
For...
======================================================
X = (equivalencies...)
------------------------
C ((k), (k-2)) ===== Binomial Notation formulation
T_(k-1) ===== n-Hedral formulation (T = Triangular Number)
(k^2 - k)/2! ===== Algebraic formulation
S [1,1;0,1]:[3,(k+1)] ===== Personal Notation (can be re-explained another time...)
Condition: k is greater than or equal to 1
Y = (equivalencies...)
------------------------
C ((k+1), (k-2)) ===== Binomial Notation formulation
TETRA_(k-1) ===== n-Hedral formulation (Tetra = Tetrahedral Number)
(k^3 - k)/3! ===== Algebraic formulation
S [1,1;0,1]:[4,(k+1)] ===== Personal Notation (can be re-explained another time...)
Condition: k is greater than or equal to 1
======================================================
QUESTIONS:
I. What are the solutions for k of X [k(x)] and k of Y [k(Y)]?
II. What are the values for n of X [n(x)] and n of Y [n(Y)]?
III. What are the values for X and Y?
IV. What are the values for Z_1 and Z_2
The possible solutions to Z_1, an eminently finite set, are already known:
k(X) = 1, 2, 3, 6, 91
n(X) = 1, 2, 3, 5, 13
X = 0, 1, 3, 15, 4095
Z_1 = 0, 2, 6, 30, 8190
The values for X are otherwise known as, yes, you possibly guessed it, the Ramanujan-Nagell Triangular Numbers.
The set of solutions for Z_2, to the best of my knowledge, is unknown, but here is what I have so far:
k(Y) = 1, 2, 3, 5, 8
n(Y) = 1, 2, 3, 5, 7
Y = 0, 1, 4, 20, 84
Z_1 = 0, 2, 6, 30, 126
Is this also a finite set?
I believe it may well be. If so, then...
{Z_1 U Z_2} = 0, 2, 6, 30, 126, 8190
In order, Unity + the first 5 Mersenne Primes, less 1.
But I could very well, of course, be mistaken. There may, in fact, be an infinite number of solutions for Z_2. Thus, my question.
I might add that, given that Nagell proved Z_1, confirming one of many Ramanujan Connjectures, it is reasonable to assume that any proofs for Z_2 would assume a similar character.
But, again, I could be wrong...
Best Regards,
Raphie
P.S. The form 2^n - 2, by the way, is related to a mathematical concept known as "Sphere Line Picking." http://mathworld.wolfram.com/SphereLinePicking.html
P.P.S. The Ramanujan-Nagell Squares are equal to 8*X + 1 = 8 * {0, 1, 3, 15, 4095} + 1 = 1, 9, 25, 121, 32761, a number set that, surprisingly, is not to be found on OEIS. Each Ramanujan-Nagell Square follows the form 2^n - 7. The indices for these square numbers are equal to (2 * k(X)) - 1) = 1, 3, 5, 11, 181.
RELATED SEQUENCES
A060728
http://www.research.att.com/~njas/sequences/A060728
Numbers n such that Ramanujan's equation x^2 + 7 = 2^n has an integer solution.
3, 4, 5, 7, 15
A038198
http://www.research.att.com/~njas/sequences/A038198
Numbers n such that n^2+7 is a power of 2.
1, 3, 5, 11, 181
QUOTE (AlphaNumeric+Aug 22 2009, 11:43 AM)
Somehow I just know that somewhere Euler is bashing his head into his keyboard in exasperation......
What I find unbelievable, is how stupid this guy must be to still not understand the uselessness of all this kiddie level numerology. I guess all we can do is pity this guy. Clearly he spends a lot of time on this drivel, and he's not quite sharp enough to realise it's all a complete waste of time. Raphie Frank: from one guy to another:
You are making a fool of yourself.
Note, I don't get any benefit by saying this. I'm not doing it for the good of my health. I have a decent knowledge of mathematics, and more importantly, I'm aware of how progress is made in mathematics. My 100% honest, no joke, completely sincere opinion of your numerology, is:
It is cringe-worthy. It involves mathematics that children do. It is written by someone who lacks mathematical talent. It is flowery: if you mean 1, say 1 - not unity. If you mean 8, say 8 - not the first prime raised to the second prime.
But I urge you - ask other mathematicians. You don't understand my explanations, that's fine. So you're left with my opinion. If you don't like it, seek the mathematical opinion of others in the form of peer review. I will continue to scan the leading mathematical journals for your contributions... unless, of course, they turn out to be the kiddie level tripe I've explained them to be!
What I find unbelievable, is how stupid this guy must be to still not understand the uselessness of all this kiddie level numerology. I guess all we can do is pity this guy. Clearly he spends a lot of time on this drivel, and he's not quite sharp enough to realise it's all a complete waste of time. Raphie Frank: from one guy to another:
You are making a fool of yourself.
Note, I don't get any benefit by saying this. I'm not doing it for the good of my health. I have a decent knowledge of mathematics, and more importantly, I'm aware of how progress is made in mathematics. My 100% honest, no joke, completely sincere opinion of your numerology, is:
It is cringe-worthy. It involves mathematics that children do. It is written by someone who lacks mathematical talent. It is flowery: if you mean 1, say 1 - not unity. If you mean 8, say 8 - not the first prime raised to the second prime.
But I urge you - ask other mathematicians. You don't understand my explanations, that's fine. So you're left with my opinion. If you don't like it, seek the mathematical opinion of others in the form of peer review. I will continue to scan the leading mathematical journals for your contributions... unless, of course, they turn out to be the kiddie level tripe I've explained them to be!
QUOTE (Raphie Frank+Aug 23 2009, 01:48 AM)
Second, if the question is "bash-worthy," there can only be two reasonable explanations why:
I didn't even read your post, I simply noted it was awash with your usual smattering of mentions of 'triangular', 'tetrahedral', 'Kissing' etc.
I didn't even read your post, I simply noted it was awash with your usual smattering of mentions of 'triangular', 'tetrahedral', 'Kissing' etc.
Instead of insulting me, I suggest either or both of you simply make me look foolish by providing the answer to the question.
It should be a piece of cake, especially, if, in fact, I am operating at the kiddie level it has been repeatedly suggested I am operating at.
Of course, Alphanumeric, you would actually have to read the post.
Best,
Raphie
P.S. To look foolish as a tradeoff for actually receiving useful (to me) knowledge is a tradeoff I would gladly make.
It should be a piece of cake, especially, if, in fact, I am operating at the kiddie level it has been repeatedly suggested I am operating at.
Of course, Alphanumeric, you would actually have to read the post.
Best,
Raphie
P.S. To look foolish as a tradeoff for actually receiving useful (to me) knowledge is a tradeoff I would gladly make.
QUOTE (Raphie Frank+Aug 23 2009, 08:32 PM)
Instead of insulting me, I suggest either or both of you simply make me look foolish by providing the answer to the question.
It should be a piece of cake, especially, if, in fact, I am operating at the kiddie level it has been repeatedly suggested I am operating at...
Would you care to illuminate me as to your purpose on this forum? So far, it seems to consist of arguing with physicists about mathematics.
It should be a piece of cake, especially, if, in fact, I am operating at the kiddie level it has been repeatedly suggested I am operating at...
Would you care to illuminate me as to your purpose on this forum? So far, it seems to consist of arguing with physicists about mathematics.
QUOTE (flyingbuttressman+Aug 24 2009, 12:36 AM)
Would you care to illuminate me as to your purpose on this forum? So far, it seems to consist of arguing with physicists about mathematics.
Although of admittedly argumentative nature, I have no desire to argue, whatsoever, within the context of this forum, flyingbuttressman. The arguments, by and large, have been forced upon me. I will address purpose in another post at some point, but only have a few minutes online at the moment. In any case, join the queue regarding unanswered questions. I still "owe" RPenner two responses and Alphanumeric one.
Best,
Raphie
Although of admittedly argumentative nature, I have no desire to argue, whatsoever, within the context of this forum, flyingbuttressman. The arguments, by and large, have been forced upon me. I will address purpose in another post at some point, but only have a few minutes online at the moment. In any case, join the queue regarding unanswered questions. I still "owe" RPenner two responses and Alphanumeric one.
Best,
Raphie
QUOTE (Euler+Aug 23 2009, 05:35 PM)
if you mean 1, say 1 - not unity. If you mean 8, say 8 - not the first prime raised to the second prime.
When I use the term Unity, it is to distinguish it from the primes and composites. UNITY is a number unto it's own, the only integer with one divisor, the only number that is One-Perfect, etc. 1, on the other hand, is just another number.
I wrote exactly, or to the best of my ability, what I meant Euler.
- RF
When I use the term Unity, it is to distinguish it from the primes and composites. UNITY is a number unto it's own, the only integer with one divisor, the only number that is One-Perfect, etc. 1, on the other hand, is just another number.
I wrote exactly, or to the best of my ability, what I meant Euler.
- RF
OMG. That is numerology. Not math.
QUOTE (flyingbuttressman+Aug 23 2009, 08:36 PM)
Would you care to illuminate me as to your purpose on this forum? So far, it seems to consist of arguing with physicists about mathematics.
Not at all, but maybe his purpose is the same as yours, American Idol!!! Of course. Go audition for it already, dahling. It's all you want, isn't it?
Go find Glory now. Maybe they will like you a bit more than we do. We don't want to fight any fights besides you. And I mean in the nicest way...
Not at all, but maybe his purpose is the same as yours, American Idol!!! Of course. Go audition for it already, dahling. It's all you want, isn't it?
Go find Glory now. Maybe they will like you a bit more than we do. We don't want to fight any fights besides you. And I mean in the nicest way...
QUOTE (rpenner+Aug 24 2009, 04:08 AM)
OMG. That is numerology. Not math.
On what basis, pray tell?
If you are referring to my comments on Unity, then I suggest you go into Wikipedia and update the record, lest others get the wrongheaded notion that 1 is different from the composites and the primes:
============================================================
One is neither a prime number nor a composite number, but a unit, like -1 and, in the Gaussian integers, i and -i. The fundamental theorem of arithmetic guarantees unique factorization over the integers only up to units (e.g. 4 = 22 = (-1)4×123×22).
http://en.wikipedia.org/wiki/1_(number)
============================================================
If, on the other hand, you are referring to the question I put to you regarding numbers of the form Ceiling (n^3 - n)/4 equal to 2^n - 2, I suggest you give folks at Wolfram and elsewhere a proper scolding for giving online exposure to such as the following...
===================================================
Ramanujan's Square Equation
In 1913, Ramanujan asked if the Diophantine equation of second order
2^n - 7 = x^2
sometimes called the Ramanujan-Nagell equation, has any solutions other than , 4, 5, 7, and 15 (Schroeppel 1972, Item 31; Ramanujan 2000, p. 327; Sloane's A060728). These correspond to , 3, 5, 11, and 181 (Sloane's A038198). Nagell (1948) and Skolem et al. (1959) showed there are no solutions past , thus establishing Ramanujan's question in the negative.
http://mathworld.wolfram.com/RamanujansSquareEquation.html
===================================================
I believe you are letting anger at me get in the way of your better judgement, RPenner.
Best,
Raphie
On what basis, pray tell?
If you are referring to my comments on Unity, then I suggest you go into Wikipedia and update the record, lest others get the wrongheaded notion that 1 is different from the composites and the primes:
============================================================
One is neither a prime number nor a composite number, but a unit, like -1 and, in the Gaussian integers, i and -i. The fundamental theorem of arithmetic guarantees unique factorization over the integers only up to units (e.g. 4 = 22 = (-1)4×123×22).
http://en.wikipedia.org/wiki/1_(number)
============================================================
If, on the other hand, you are referring to the question I put to you regarding numbers of the form Ceiling (n^3 - n)/4 equal to 2^n - 2, I suggest you give folks at Wolfram and elsewhere a proper scolding for giving online exposure to such as the following...
===================================================
Ramanujan's Square Equation
In 1913, Ramanujan asked if the Diophantine equation of second order
2^n - 7 = x^2
sometimes called the Ramanujan-Nagell equation, has any solutions other than , 4, 5, 7, and 15 (Schroeppel 1972, Item 31; Ramanujan 2000, p. 327; Sloane's A060728). These correspond to , 3, 5, 11, and 181 (Sloane's A038198). Nagell (1948) and Skolem et al. (1959) showed there are no solutions past , thus establishing Ramanujan's question in the negative.
http://mathworld.wolfram.com/RamanujansSquareEquation.html
===================================================
I believe you are letting anger at me get in the way of your better judgement, RPenner.
Best,
Raphie
QUOTE (Raphie Frank+Aug 24 2009, 12:32 AM)
Instead of insulting me, I suggest either or both of you simply make me look foolish by providing the answer to the question.
You utter, utter ignoramus. You see, these type of comments just help to illustrate the over-inflated opinion you have of your numerology. You can't even construct a mathematical question!
But let me elaborate - here is your "question":
You utter, utter ignoramus. You see, these type of comments just help to illustrate the over-inflated opinion you have of your numerology. You can't even construct a mathematical question!
But let me elaborate - here is your "question":
QUOTE (Raphie Frank+Aug 23 2009, 12:48 AM)
Let...
Z_1 = Ceiling (2*X) = 2^n - 2 Is Z_1 a function?
Z_2 = Ceiling (1.5*Y) = 2^n - 2 Is Z_2 a function?
For...
======================================================
X = (equivalencies...) What? I mean seriously, what?
------------------------
C ((k), (k-2)) ===== Binomial Notation formulation So you mean k(k-1)/2. What is k?
T_(k-1) ===== n-Hedral formulation (T = Triangular Number) So you mean k(k-1)/2 again? What is k?
(k^2 - k)/2! ===== Algebraic formulation Again, you mean k(k-1)/2. You've written the same thing three times. Here's another way [(k-1/2)^2 -1/4]/2. Wow - that was deep.
S [1,1;0,1]:[3,(k+1)] ===== Personal Notation (can be re-explained another time...) So this has, quite literally, no meaning to anyone but yourself? Why include it - oh, I remember, you like to write as much as possible. As in the above - three flowery definitions of the expression "k(k-1)/2".
Condition: k is greater than or equal to 1
From this, I can only guess that all that fluff means X is a function on the positive integers and is very simple X(k) = k(k-1)/2
Y = (equivalencies...) Again, what?
------------------------
C ((k+1), (k-2)) ===== Binomial Notation formulation So you mean (k^3-k)/6. What is k?
TETRA_(k-1) ===== n-Hedral formulation (Tetra = Tetrahedral Number) So again you mean (k^3-k)/6. Surprise!
(k^3 - k)/3! ===== Algebraic formulation Third time you've written the same thing.
S [1,1;0,1]:[4,(k+1)] ===== Personal Notation (can be re-explained another time...) Again, this adds nothing. One can only conclude the author wanted it included because he feels it looks good. Stupid.
Condition: k is greater than or equal to 1
From this, I can only guess that all that fluff means Y is function on the positive integers and is very simple Y(k) = (k^3 - k)/6
======================================================
QUESTIONS:
I. What are the solutions for k of X [k(x)] and k of Y [k(Y)]? You have written no equations, so what solutions are you on about?
II. What are the values for n of X [n(x)] and n of Y [n(Y)]? This makes no sense. Author does not even realise this.
III. What are the values for X and Y? This makes no sense! X and Y are functions!
IV. What are the values for Z_1 and Z_2 This makes no sense! Z_1 and Z_2 are functions!
You stupid, confused little man.
Z_1 = Ceiling (2*X) = 2^n - 2 Is Z_1 a function?
Z_2 = Ceiling (1.5*Y) = 2^n - 2 Is Z_2 a function?
For...
======================================================
X = (equivalencies...) What? I mean seriously, what?
------------------------
C ((k), (k-2)) ===== Binomial Notation formulation So you mean k(k-1)/2. What is k?
T_(k-1) ===== n-Hedral formulation (T = Triangular Number) So you mean k(k-1)/2 again? What is k?
(k^2 - k)/2! ===== Algebraic formulation Again, you mean k(k-1)/2. You've written the same thing three times. Here's another way [(k-1/2)^2 -1/4]/2. Wow - that was deep.
S [1,1;0,1]:[3,(k+1)] ===== Personal Notation (can be re-explained another time...) So this has, quite literally, no meaning to anyone but yourself? Why include it - oh, I remember, you like to write as much as possible. As in the above - three flowery definitions of the expression "k(k-1)/2".
Condition: k is greater than or equal to 1
From this, I can only guess that all that fluff means X is a function on the positive integers and is very simple X(k) = k(k-1)/2
Y = (equivalencies...) Again, what?
------------------------
C ((k+1), (k-2)) ===== Binomial Notation formulation So you mean (k^3-k)/6. What is k?
TETRA_(k-1) ===== n-Hedral formulation (Tetra = Tetrahedral Number) So again you mean (k^3-k)/6. Surprise!
(k^3 - k)/3! ===== Algebraic formulation Third time you've written the same thing.
S [1,1;0,1]:[4,(k+1)] ===== Personal Notation (can be re-explained another time...) Again, this adds nothing. One can only conclude the author wanted it included because he feels it looks good. Stupid.
Condition: k is greater than or equal to 1
From this, I can only guess that all that fluff means Y is function on the positive integers and is very simple Y(k) = (k^3 - k)/6
======================================================
QUESTIONS:
I. What are the solutions for k of X [k(x)] and k of Y [k(Y)]? You have written no equations, so what solutions are you on about?
II. What are the values for n of X [n(x)] and n of Y [n(Y)]? This makes no sense. Author does not even realise this.
III. What are the values for X and Y? This makes no sense! X and Y are functions!
IV. What are the values for Z_1 and Z_2 This makes no sense! Z_1 and Z_2 are functions!
You stupid, confused little man.
QUOTE (Euler+Aug 24 2009, 07:43 AM)
You can't even construct a mathematical question!
Co to je matematika, Euler?
The language may be different, but the question is the same...
What is mathematics?
Which is more important? The specific language or the idea?
The fact is that my language is quite clear, Euler, but it seems you have been trained into a very toadish and limited understanding of mathematics, confusing the description of the "thing" for the "thing" itself to the point that you can not understand basic statements a Junior High Schooler could understand.
This is the unacknowledged bane of "experts" just about everywhere.
For the record, as with Alphanumeric, I did not even bother to read your post past the first sentence and the last sentence. Your vitriol and childishness made it well not worth my time.
I am hopeful that, on the basis of this, you will charge me with making statements based upon ignorance so that I may then quote your words back to Alphanumeric.
Best,
Raphie
@ flyingbuttressman
Just who is creating the arguments here? Shall I stand by and accept the quite rather demented judgements of very "little men" without response or defense?
Co to je matematika, Euler?
The language may be different, but the question is the same...
What is mathematics?
Which is more important? The specific language or the idea?
The fact is that my language is quite clear, Euler, but it seems you have been trained into a very toadish and limited understanding of mathematics, confusing the description of the "thing" for the "thing" itself to the point that you can not understand basic statements a Junior High Schooler could understand.
This is the unacknowledged bane of "experts" just about everywhere.
For the record, as with Alphanumeric, I did not even bother to read your post past the first sentence and the last sentence. Your vitriol and childishness made it well not worth my time.
I am hopeful that, on the basis of this, you will charge me with making statements based upon ignorance so that I may then quote your words back to Alphanumeric.
Best,
Raphie
@ flyingbuttressman
Just who is creating the arguments here? Shall I stand by and accept the quite rather demented judgements of very "little men" without response or defense?
QUOTE (Raphie Frank+Aug 24 2009, 09:32 AM)
I did not even bother to read your post past the first sentence
Ahh, yes. Best to stick your head in the sand!
Ahh, yes. Best to stick your head in the sand!
QUOTE (Euler+Aug 24 2009, 07:43 AM)
But let me elaborate - here is your "question":
Having now read past the first sentence...
===============================
No, Euler, HERE is my question:
[... and let me make it so simple and clear that even close-minded, bigoted (pardon my asterisked French...) dumbfu*ks and the moderators who, via malevolent or benevolent neglect (the reader may choose....), enable them, might understand...]
ARE THERE ANY VALUES FOR x BEYOND x = 8, SUCH THAT Ceiling (x^3 - x)/4 = 2^n - 2?
- RF
Having now read past the first sentence...
===============================
No, Euler, HERE is my question:
[... and let me make it so simple and clear that even close-minded, bigoted (pardon my asterisked French...) dumbfu*ks and the moderators who, via malevolent or benevolent neglect (the reader may choose....), enable them, might understand...]
ARE THERE ANY VALUES FOR x BEYOND x = 8, SUCH THAT Ceiling (x^3 - x)/4 = 2^n - 2?
- RF
QUOTE (Raphie Frank+Aug 24 2009, 09:41 AM)
Having now read past the first sentence...
===============================
No, Euler, HERE is my question:
[... and let me make it so simple and clear that even close-minded, bigoted (pardon my asterisked French...) dumbfu*ks and the moderators who, via malevolent or benevolent neglect (the reader may choose....), enable them, might understand...]
ARE THERE ANY VALUES FOR x BEYOND x = 8, SUCH THAT Ceiling (x^3 - x)/4 = 2^n - 2?
- RF
Ha! Which just serves to prove my point. You had a question that is unbelievably simple to state, but you attempted to dress it up so much that it became meaningless. A prime example of your empty, flowery posts.
Good job.
===============================
No, Euler, HERE is my question:
[... and let me make it so simple and clear that even close-minded, bigoted (pardon my asterisked French...) dumbfu*ks and the moderators who, via malevolent or benevolent neglect (the reader may choose....), enable them, might understand...]
ARE THERE ANY VALUES FOR x BEYOND x = 8, SUCH THAT Ceiling (x^3 - x)/4 = 2^n - 2?
- RF
Ha! Which just serves to prove my point. You had a question that is unbelievably simple to state, but you attempted to dress it up so much that it became meaningless. A prime example of your empty, flowery posts.
Good job.
QUOTE (Euler+Aug 24 2009, 09:46 AM)
Ha! Which just serves to prove my point. You had a question that is unbelievably simple to state, but you attempted to dress it up so much that it became meaningless.
Fair point Euler, although there was no attempt to "dress up," but rather an attempt to provide context. Now answer the question or go away.
Best,
Raphie
Fair point Euler, although there was no attempt to "dress up," but rather an attempt to provide context. Now answer the question or go away.
Best,
Raphie
QUOTE (Raphie Frank+Aug 24 2009, 09:49 AM)
Fair point Euler, although there was no attempt to "dress up," but rather an attempt to provide context. Now answer the question or go away.
Rubbish - it was your classic attempt to make things sound more complex than they are. To answer your question: there are lots - try x=10.08 for example.
Rubbish - it was your classic attempt to make things sound more complex than they are. To answer your question: there are lots - try x=10.08 for example.
QUOTE (Euler+Aug 24 2009, 10:57 AM)
try x=10.08 for example.
We are dealing with integral solutions only, Euler, so, sorry, no go.
You'll have to excuse me if I assumed this to be clear given the obviously stated reference to the Ramanujan-Nagell equation, Diophantine in nature, much less the EXPLICITLY stated "ceiling" condition.
- RF
We are dealing with integral solutions only, Euler, so, sorry, no go.
You'll have to excuse me if I assumed this to be clear given the obviously stated reference to the Ramanujan-Nagell equation, Diophantine in nature, much less the EXPLICITLY stated "ceiling" condition.
- RF
QUOTE (Raphie Frank+Aug 24 2009, 11:05 AM)
We are dealing with integral solutions only, Euler, so, sorry, no go.
You'll have to excuse me if I assumed this to be clear given the obviously stated reference to the Ramanujan-Nagell equation, Diophantine in nature, much less the EXPLICITLY stated "ceiling" condition.
- RF
No where have you stated x needed to be integral. What is more, your "ceiling" comment suggests you don't realise that the function is perfectly well defined for non integral x.
Nice job, again.
You'll have to excuse me if I assumed this to be clear given the obviously stated reference to the Ramanujan-Nagell equation, Diophantine in nature, much less the EXPLICITLY stated "ceiling" condition.
- RF
No where have you stated x needed to be integral. What is more, your "ceiling" comment suggests you don't realise that the function is perfectly well defined for non integral x.
Nice job, again.
QUOTE (Euler+Aug 24 2009, 11:08 AM)
No where have you stated x needed to be integral. What is more, your "ceiling" comment suggests you don't realise that the function is perfectly well defined for non integral x.
Nice job, again.
Note to the general reader regarding (IMO) misleading information being provided by "experts":
CEILING (x), whatever x may be, typically refers to the closest integer above, while FLOOR typically refers to the closest integer below. In the context of this thread, by way of example...
==============================================================
A118841
Numbers n such that Ceiling[GoldenRatio^n] is prime.
1, 2, 3, 4, 8, 16
COMMENT
a(7)>3000 or non-existent. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 07 2007
http://www.research.att.com/~njas/sequences/A118841
==============================================================
Observe that there is no mention of integral solutions, specifically, as this is implied by way of standard usage, although I've no doubt that the "function is perfectly well defined for non integral x." Furthermore...
==============================================================
Floor and ceiling functions
via Wikipedia, the free encyclopedia
In mathematics and computer science, the floor and ceiling functions map a real number to the next smallest or next largest integer. More precisely, floor(x) is the largest integer not greater than x and ceiling(x) is the smallest integer not less than x.[1]
http://en.wikipedia.org/wiki/Floor_and_ceiling_functions
==============================================================
Best,
Raphie
P.S. Rpenner has already shown upon this thread, by the way, that the statement "a(7)>3000 or non-existent" can be revised upwards.
Nice job, again.
Note to the general reader regarding (IMO) misleading information being provided by "experts":
CEILING (x), whatever x may be, typically refers to the closest integer above, while FLOOR typically refers to the closest integer below. In the context of this thread, by way of example...
==============================================================
A118841
Numbers n such that Ceiling[GoldenRatio^n] is prime.
1, 2, 3, 4, 8, 16
COMMENT
a(7)>3000 or non-existent. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 07 2007
http://www.research.att.com/~njas/sequences/A118841
==============================================================
Observe that there is no mention of integral solutions, specifically, as this is implied by way of standard usage, although I've no doubt that the "function is perfectly well defined for non integral x." Furthermore...
==============================================================
Floor and ceiling functions
via Wikipedia, the free encyclopedia
In mathematics and computer science, the floor and ceiling functions map a real number to the next smallest or next largest integer. More precisely, floor(x) is the largest integer not greater than x and ceiling(x) is the smallest integer not less than x.[1]
http://en.wikipedia.org/wiki/Floor_and_ceiling_functions
==============================================================
Best,
Raphie
P.S. Rpenner has already shown upon this thread, by the way, that the statement "a(7)>3000 or non-existent" can be revised upwards.
QUOTE (Raphie Frank+Aug 25 2009, 05:43 AM)
Note to the general reader regarding (IMO) misleading information being provided by "experts":
CEILING (x), whatever x may be, typically refers to the closest integer above, while FLOOR typically refers to the closest integer below. In the context of this thread, by way of example...
What are you on about? How utterly inept can you possibly be? Your reference to "ceiling functions" makes absolutely no difference to the validity of the solution I provided. In fact, the presence of ceiling functions actually indicates that the author is interested in non-integral solutions, but you don't realise this because you have the mathematical talent of a dead sheep.
If you were only looking for integer solutions, then it should be obvious that one can remove the "ceiling" function from the equation, given two seconds thought. Clearly you didn't realise this, which indicates a staggering lack of mathematical competence (even for a, what was it, number researcher? Chuckle...)
Come on, Mr. Thicky. Show us all what you're made of. If you're only interested in integer x's, then use this information to quickly eliminate the need for ceiling functions. I will even tell you how to do it: work mod 8, there are two cases of interest.
CEILING (x), whatever x may be, typically refers to the closest integer above, while FLOOR typically refers to the closest integer below. In the context of this thread, by way of example...
What are you on about? How utterly inept can you possibly be? Your reference to "ceiling functions" makes absolutely no difference to the validity of the solution I provided. In fact, the presence of ceiling functions actually indicates that the author is interested in non-integral solutions, but you don't realise this because you have the mathematical talent of a dead sheep.
If you were only looking for integer solutions, then it should be obvious that one can remove the "ceiling" function from the equation, given two seconds thought. Clearly you didn't realise this, which indicates a staggering lack of mathematical competence (even for a, what was it, number researcher? Chuckle...)
Come on, Mr. Thicky. Show us all what you're made of. If you're only interested in integer x's, then use this information to quickly eliminate the need for ceiling functions. I will even tell you how to do it: work mod 8, there are two cases of interest.
QUOTE (Raphie Frank+Aug 25 2009, 06:43 AM)
Note to the general reader regarding (IMO) misleading information being provided by "experts":
CEILING (x), whatever x may be, typically refers to the closest integer above, while FLOOR typically refers to the closest integer below. In the context of this thread, by way of example...
If x is an integer than Ceiling(x) = x and Floor(x) = x. In which case your question reduces to having not ceiling or floor functions. The whole point of having those functions is so that you can allow for values of x which are not integers. Given the nature of those functions, if you find an integer solution you find infinitely many solutions because you can add any number less than 1 and still get the same result.
If you're working with integers then you cannot have any ceiling or floor functions applied to x (or whatever your unknown to be found is) else you allow non-integer solutions.
For instance
3x = 6. Integer solution only, x=2
3Floor(x) = 6. Interval of solutions 2=<x<3.
Understand? Or shall I be like you and ask that in 3 different languages? Whatever language I say it in you're still an idiot.
CEILING (x), whatever x may be, typically refers to the closest integer above, while FLOOR typically refers to the closest integer below. In the context of this thread, by way of example...
If x is an integer than Ceiling(x) = x and Floor(x) = x. In which case your question reduces to having not ceiling or floor functions. The whole point of having those functions is so that you can allow for values of x which are not integers. Given the nature of those functions, if you find an integer solution you find infinitely many solutions because you can add any number less than 1 and still get the same result.
If you're working with integers then you cannot have any ceiling or floor functions applied to x (or whatever your unknown to be found is) else you allow non-integer solutions.
For instance
3x = 6. Integer solution only, x=2
3Floor(x) = 6. Interval of solutions 2=<x<3.
Understand? Or shall I be like you and ask that in 3 different languages? Whatever language I say it in you're still an idiot.
QUOTE (AlphaNumeric+Aug 25 2009, 08:47 AM)
If x is an integer than Ceiling(x) = x and Floor(x) = x. In which case your question reduces to having not ceiling or floor functions.
I'm assuming, when he writes Ceiling (x^3 - x) / 4, he means Ceiling [(x^3 - x)/4]. But I can't be sure - he's pretty dense after all. But even if it is the latter case, considering (x^3-x) mod 8 you can quickly remove the ceiling function: (x^3-x) is obviously divisible by 2, so only need to consider (x^3-x)=0,2,4,6 mod 8. Using [ ] to denote ceiling function:
0 mod 8: [(x^3-x)/4] = 2a (even, good)
2 mod 8: [(x^3-x)/4] = 2a + [2/4] = 2a + 1 (odd, bad)
4 mod 8: [(x^3-x)/4] = 2a + [4/4] = 2a + 1 (odd, bad)
6 mod 8: [(x^3-x)/4] = 2a + [6/4] = 2a + 2 (even, good)
(good/bad since, from the equation, [(x^3-x)/4] is even). So there are two cases to consider. But it would seem Raphie Frank, the famous "number researcher", with diplomas in creative writing, didn't manage to spot this. What is the world coming to...
I'm assuming, when he writes Ceiling (x^3 - x) / 4, he means Ceiling [(x^3 - x)/4]. But I can't be sure - he's pretty dense after all. But even if it is the latter case, considering (x^3-x) mod 8 you can quickly remove the ceiling function: (x^3-x) is obviously divisible by 2, so only need to consider (x^3-x)=0,2,4,6 mod 8. Using [ ] to denote ceiling function:
0 mod 8: [(x^3-x)/4] = 2a (even, good)
2 mod 8: [(x^3-x)/4] = 2a + [2/4] = 2a + 1 (odd, bad)
4 mod 8: [(x^3-x)/4] = 2a + [4/4] = 2a + 1 (odd, bad)
6 mod 8: [(x^3-x)/4] = 2a + [6/4] = 2a + 2 (even, good)
(good/bad since, from the equation, [(x^3-x)/4] is even). So there are two cases to consider. But it would seem Raphie Frank, the famous "number researcher", with diplomas in creative writing, didn't manage to spot this. What is the world coming to...
Utterly surreal...
@ flyingbuttressman
In reference to the ad hominem nature of argumentation upon this thread...
[edit\]I will let other posters words speak for themselves... But this much I will state... do you not recognize at all the irony in a scenario where mathematicans and/or physicists are so hell bent on discrediting another than they are even willing to provide arguments AGAINST the standard usage of accepted mathematical terminology?[/edit]
@ flyingbuttressman
In reference to the ad hominem nature of argumentation upon this thread...
[edit\]I will let other posters words speak for themselves... But this much I will state... do you not recognize at all the irony in a scenario where mathematicans and/or physicists are so hell bent on discrediting another than they are even willing to provide arguments AGAINST the standard usage of accepted mathematical terminology?[/edit]
Ahh, bless. He's back to sticking his head in the sand.
QUOTE (AlphaNumeric+Aug 25 2009, 08:47 AM)
If you're working with integers then you cannot have any ceiling or floor functions applied to x (or whatever your unknown to be found is) else you allow non-integer solutions.
If one's interest is Kissing Numbers, then ceiling and floor are not only "can," but "must."
If such functions were not applicable, then the Dimension 3 Kissing Number would be 13, not 12, as David Gregory wrongly argued, and Isaac Newton correctly argued, over 300 years ago.
Oh, but wait, I forgot, I am addressing one who yet believes, or at least has argued, that Isaac Newton broke no accepted rules OF THE DAY, in developing calculus. My mistake. For a moment there I believed myself to be dealing with someone logically, rather than emotionally, driven...
But let me humour you and we can remove the ceiling function, but EXPLICITLY specify, even for the brain dead, that we are limiting ourselves to integral solutions (where x is greater than or equal to 0) to the question. (Note to the reader: "Integral" meaning WHOLE NUMBER)
Clear enough to understand?
Or, perhaps, you understand the question already , but are merely seeking points to contend with?
- RF
Note to RPenner:
Time for you to launch a impeccably timed one line innuendo? Don't worry about substantiation. Such is only a requirement for those on the receiving end of your critiques... Might I suggest that you redline this post, as well?
If one's interest is Kissing Numbers, then ceiling and floor are not only "can," but "must."
If such functions were not applicable, then the Dimension 3 Kissing Number would be 13, not 12, as David Gregory wrongly argued, and Isaac Newton correctly argued, over 300 years ago.
Oh, but wait, I forgot, I am addressing one who yet believes, or at least has argued, that Isaac Newton broke no accepted rules OF THE DAY, in developing calculus. My mistake. For a moment there I believed myself to be dealing with someone logically, rather than emotionally, driven...
But let me humour you and we can remove the ceiling function, but EXPLICITLY specify, even for the brain dead, that we are limiting ourselves to integral solutions (where x is greater than or equal to 0) to the question. (Note to the reader: "Integral" meaning WHOLE NUMBER)
Clear enough to understand?
Or, perhaps, you understand the question already , but are merely seeking points to contend with?
- RF
Note to RPenner:
Time for you to launch a impeccably timed one line innuendo? Don't worry about substantiation. Such is only a requirement for those on the receiving end of your critiques... Might I suggest that you redline this post, as well?
QUOTE (Raphie Frank+Aug 25 2009, 10:31 AM)
If one's interest is Kissing Numbers, then ceiling and floor are not only "can," but "must."
Hmmmm, I don't know how many times people have to tell you, but you're not doing any mathematics that relates to the kissing numbers. You've just picked a small collection of numbers to play numerology with, and these numbers happen to coincide with some numbers that appear in a problem in geometry.
If I wanted to play numerology with the numbers (1,3,3,1,0), I'd be doing exactly that, playing numerology. The fact that these numbers arise in the topological classification of a 3-torus doesn't make a blind bit of difference.
Aside from all that, which we've already told you. I'm beginning to worry: you haven't responded to the mathematical points raised, which might lead me to the scary conclusion that you're avoiding them because you're confused. Don't let this be the case, Raphie Frank - come show off your number-researcher skills!
Hmmmm, I don't know how many times people have to tell you, but you're not doing any mathematics that relates to the kissing numbers. You've just picked a small collection of numbers to play numerology with, and these numbers happen to coincide with some numbers that appear in a problem in geometry.
If I wanted to play numerology with the numbers (1,3,3,1,0), I'd be doing exactly that, playing numerology. The fact that these numbers arise in the topological classification of a 3-torus doesn't make a blind bit of difference.
Aside from all that, which we've already told you. I'm beginning to worry: you haven't responded to the mathematical points raised, which might lead me to the scary conclusion that you're avoiding them because you're confused. Don't let this be the case, Raphie Frank - come show off your number-researcher skills!
You are changing the subject, Euler.
The question ought to be a piece of cake for you to answer... so answer it.
Provide the counter-example.
ARE THERE ANY VALUES FOR x BEYOND x = 8, SUCH THAT Ceiling (x^3 - x)/4 = 2^n - 2?
(feel free to remove the ceiling requirement, as long as we limit ourselves to integral x's greater than or equal to 0)
My GUESS? And it is just this... a (well-reasoned) guess
You won't be able to provide a counter-example (although I would happily be proven wrong...)
Better would be to show that the question is essentially one and the same (in form) as the Ramanujan Nagell Equation.
This, however, is a tack I can count on you not to take, as your seeming "purpose" here has nothing to do with the advancement of mathematical or human knowledge...
Best,
RF
The question ought to be a piece of cake for you to answer... so answer it.
Provide the counter-example.
ARE THERE ANY VALUES FOR x BEYOND x = 8, SUCH THAT Ceiling (x^3 - x)/4 = 2^n - 2?
(feel free to remove the ceiling requirement, as long as we limit ourselves to integral x's greater than or equal to 0)
My GUESS? And it is just this... a (well-reasoned) guess
You won't be able to provide a counter-example (although I would happily be proven wrong...)
Better would be to show that the question is essentially one and the same (in form) as the Ramanujan Nagell Equation.
This, however, is a tack I can count on you not to take, as your seeming "purpose" here has nothing to do with the advancement of mathematical or human knowledge...
Best,
RF
QUOTE (Raphie Frank+Aug 25 2009, 11:04 AM)
You are changing the subject, Euler.
The question ought to be a piece of cake for you to answer... so answer it.
Provide the counter-example.
My GUESS? And it is just this... a (well-reasoned) guess
You won't be able to (although I would happily be proven wrong...
Why should it be a piece of cake? This indicates you have little-to-no knowledge of number theoretical problems. I imagine the problem is exceptionally hard. But you wouldn't understand this, because you're intellectually and mathematically unarmed.
The question ought to be a piece of cake for you to answer... so answer it.
Provide the counter-example.
My GUESS? And it is just this... a (well-reasoned) guess
You won't be able to (although I would happily be proven wrong...
Why should it be a piece of cake? This indicates you have little-to-no knowledge of number theoretical problems. I imagine the problem is exceptionally hard. But you wouldn't understand this, because you're intellectually and mathematically unarmed.
QUOTE (Raphie Frank+Aug 25 2009, 11:04 AM)
Better would be to show that the question is essentially one and the same (in form) as the Ramanujan Nagell Equation.
Ermm, I've just told you how to do this. Are you that mathematically inept? Jesus - you can't even understand answers when they're given to you! No wonder you didn't respond to any of the mathematics - it's all completely over your head!!
Ermm, I've just told you how to do this. Are you that mathematically inept? Jesus - you can't even understand answers when they're given to you! No wonder you didn't respond to any of the mathematics - it's all completely over your head!!
QUOTE (Raphie Frank+Aug 25 2009, 11:04 AM)
This, however, is a tack I can count on you not to take, as your seeming "purpose" here has nothing to do with the advancement of mathematical or human knowledge...
Wow - you class what you do as in someway "advancing human knowledge"? What an idiot!
EDIT: And with ~2 lines of code: there are no integer solutions for 8<x<100,000,000. This doesn't tell us much though. I suspect a proof is very, very difficult.
Wow - you class what you do as in someway "advancing human knowledge"? What an idiot!
EDIT: And with ~2 lines of code: there are no integer solutions for 8<x<100,000,000. This doesn't tell us much though. I suspect a proof is very, very difficult.
QUOTE (Euler+Aug 25 2009, 11:10 AM)
I imagine the problem is exceptionally hard. But you wouldn't understand this, because you're intellectually and mathematically unarmed.
I have given you a wonderful "starter position" by which to solve the problem. Nagell's proof, which I have not read and would likely not understand, but you would.
You are correct that I am "unarmed" from a mathematical "toolbox" perspective, but from the mental toolbox perspective, you'd have to explain how I rather "randomly" offered up a speculation you just confirmed to... how many numbers was that again?
100,000,000?
Your post seems to have been edited somewhere along the way...
For shame... And you wonder why I suggest you to be against mathematical knowledge or advancement?
Best,
Raphie
I have given you a wonderful "starter position" by which to solve the problem. Nagell's proof, which I have not read and would likely not understand, but you would.
You are correct that I am "unarmed" from a mathematical "toolbox" perspective, but from the mental toolbox perspective, you'd have to explain how I rather "randomly" offered up a speculation you just confirmed to... how many numbers was that again?
100,000,000?
Your post seems to have been edited somewhere along the way...
For shame... And you wonder why I suggest you to be against mathematical knowledge or advancement?
Best,
Raphie
QUOTE (Raphie Frank+Aug 25 2009, 11:22 AM)
I have given you a wonderful "starter position" by which to solve the problem. Nagell's proof, which I have not read and would likely not understand, but you would.
How on earth can you judge the worth of the proof to another result? You don't understand the proof, so you have no idea as to its applicable range. It is almost a complete certainty that the proof will not generalise, or it would have been done.
How on earth can you judge the worth of the proof to another result? You don't understand the proof, so you have no idea as to its applicable range. It is almost a complete certainty that the proof will not generalise, or it would have been done.
QUOTE (Raphie Frank+Aug 25 2009, 11:22 AM)
You are correct that I am "unarmed" from a mathematical "toolbox" perspective, but from the mental toolbox perspective, you'd have to explain how I rather "randomly" offered up a speculation you just confirmed to... how many numbers was that again?
There is no doubt, whatsoever, that you are completely intellectually unarmed. Give a child the equation x^n+y^n=z^n, with n>2, and leave them with it for an hour. They'll guess it can't be solved - wow, huge achievement. Can they prove it? No.
Guesses mean nothing. But because you don't understand anything more than blind numerology, you don't understand this.
There is no doubt, whatsoever, that you are completely intellectually unarmed. Give a child the equation x^n+y^n=z^n, with n>2, and leave them with it for an hour. They'll guess it can't be solved - wow, huge achievement. Can they prove it? No.
Guesses mean nothing. But because you don't understand anything more than blind numerology, you don't understand this.
QUOTE (Raphie Frank+Aug 25 2009, 11:22 AM)
And you wonder why I suggest you to be against mathematical knowledge or advancement?
Sorry, what was that? Which of us spends his days playing numerology on internet forums, and which of us regularly publishes results in world leading mathematical journals? Seriously, do you not think about what you're typing, even for a second?
And I'm assuming, by your continued refusal to address any of the mathematics that has been pointed out to you, that you don't understand it. Nice job, again.
Sorry, what was that? Which of us spends his days playing numerology on internet forums, and which of us regularly publishes results in world leading mathematical journals? Seriously, do you not think about what you're typing, even for a second?
And I'm assuming, by your continued refusal to address any of the mathematics that has been pointed out to you, that you don't understand it. Nice job, again.
QUOTE (Euler+Aug 25 2009, 11:34 AM)
Which of us spends his days playing numerology on internet forums, and which of us regularly publishes results in world leading mathematical journals?
And which one of us has been published ABOUT?
There are those who write papers and those who do the things that others write papers (and books) about... I qualify for the latter. You for the former. I will not judge merit.
Although, for the record, I conducted 50 interviews for a top 100 blog and my photography has appeared in both Time Out London and the Evening Standard. Not mathematics to be sure, but I am indeed "published," so to speak...
- RF
And which one of us has been published ABOUT?
There are those who write papers and those who do the things that others write papers (and books) about... I qualify for the latter. You for the former. I will not judge merit.
Although, for the record, I conducted 50 interviews for a top 100 blog and my photography has appeared in both Time Out London and the Evening Standard. Not mathematics to be sure, but I am indeed "published," so to speak...
- RF
QUOTE (Raphie Frank+Aug 25 2009, 11:40 AM)
And which one of us has been published ABOUT?
There are those who write papers and those who do the things that others write papers (and books) about... I qualify for the latter. You for the former. I will not judge merit.
Wow. Just wow.
In a thread in which someone questions your intellectual and mathematical capabilities, you counter these points with "someone wrote an article on me, and I've interviewed people about blogs". It's as if you want the readers to think you're stupid.
There are those who write papers and those who do the things that others write papers (and books) about... I qualify for the latter. You for the former. I will not judge merit.
Wow. Just wow.
In a thread in which someone questions your intellectual and mathematical capabilities, you counter these points with "someone wrote an article on me, and I've interviewed people about blogs". It's as if you want the readers to think you're stupid.
QUOTE (Raphie Frank+Aug 25 2009, 12:40 PM)
Although, for the record, I conducted 50 interviews for a top 100 blog and my photography has appeared in both Time Out London and the Evening Standard. Not mathematics to be sure, but I am indeed "published," so to speak...
And I've been in the same room as Witten. Anyone want my autograph? And this one time I held a door open for a Fields Medalist. It was awesome.
And I've been in the same room as Witten. Anyone want my autograph? And this one time I held a door open for a Fields Medalist. It was awesome.
While we're at it, I've spoken to Hawking. He didn't speak to me though, I was most insulted! 
I also won some tennis tournaments when I was younger, and there were several articles in the local news. I'm that special.
So, uh.. wait. To be "published" is worthy of bragging rights in mathematics, but not other fields? To the point that no less than 3 posters commented upon this in a spirit of ridicule within the span of 1/2 hour?
Much less, that such comments follow within another half hour or so of this board's primary critiquer of Raphie Frank confirming to 100,000,000 that no integer values satisfy the formula (n^3 - n)/4 = 2^k - 2, the question that is the secondary topic of this thread?
I would say "call me nutty, but..." but this has already been done many times.
As such, I will simply reiterate my previous statement:
"Utterly surreal"
Cheers,
Raphie
Much less, that such comments follow within another half hour or so of this board's primary critiquer of Raphie Frank confirming to 100,000,000 that no integer values satisfy the formula (n^3 - n)/4 = 2^k - 2, the question that is the secondary topic of this thread?
I would say "call me nutty, but..." but this has already been done many times.
As such, I will simply reiterate my previous statement:
"Utterly surreal"
Cheers,
Raphie
QUOTE (Raphie Frank+Aug 25 2009, 12:22 PM)
So, uh.. wait. To be "published" is worthy of bragging rights in mathematics, but not other fields? To the point that no less than 3 posters commented upon this in a spirit of ridicule within the span of 1/2 hour?
Clearly there's a language barrier. When people who produce actual academic research use the term "publish", it is understood to mean "publish a research article in an academic journal". Not "I wrote a letter to Paperclips-Monthly, and they published it".
I can only assume "number researchers" allow for the second usage, in their informal definition.
Clearly there's a language barrier. When people who produce actual academic research use the term "publish", it is understood to mean "publish a research article in an academic journal". Not "I wrote a letter to Paperclips-Monthly, and they published it".
I can only assume "number researchers" allow for the second usage, in their informal definition.
QUOTE (Raphie Frank+Aug 25 2009, 12:22 PM)
Much less, that such comments follow within another half hour or so of this board's primary critiquer of Raphie Frank confirming to 100,000,000 that no integer values satisfy the formula (n^3 - n)/4 = 2^n - 2, the question that is the secondary topic of this thread?
Erm, hold on - where has the ceiling function gone? Why have you just dropped it? That can't be right, since (for example) for n=2, (n^3-n)/4 = 3/2 != ceiling (3/2). Oh dear - is somebody confused again?
But now you've changed the equation. It is easy to prove that the only solutions to your new equation are n=0,3,5.
Is all this getting too much for you?
Erm, hold on - where has the ceiling function gone? Why have you just dropped it? That can't be right, since (for example) for n=2, (n^3-n)/4 = 3/2 != ceiling (3/2). Oh dear - is somebody confused again?
But now you've changed the equation. It is easy to prove that the only solutions to your new equation are n=0,3,5.
Is all this getting too much for you?
QUOTE (Euler+Aug 25 2009, 12:37 PM)
Not "I wrote a letter to Paperclips-Monthly, and they published it".
Umm... Gothamist.com, named Top City Blog by Forbes Magazine, The Evening Standard (of London... over half a page in the "Going Out" section...), Time Out London...
These, I guess, are NOTHING next to "Paperclips Monthly." Perhaps I will one day be so lucky as to be published therein???
I can only squeeze my hands together and fervently pray that I should be so blessed.
All the same, I am far more proud of partaking in events (actually initiating or reinitiating them...) that have won others awards (i.e Ray Parker at Vassar) and helped launch careers (i.e Peter Dubois - Theater Director.)
- RF
Umm... Gothamist.com, named Top City Blog by Forbes Magazine, The Evening Standard (of London... over half a page in the "Going Out" section...), Time Out London...
These, I guess, are NOTHING next to "Paperclips Monthly." Perhaps I will one day be so lucky as to be published therein???
I can only squeeze my hands together and fervently pray that I should be so blessed.
All the same, I am far more proud of partaking in events (actually initiating or reinitiating them...) that have won others awards (i.e Ray Parker at Vassar) and helped launch careers (i.e Peter Dubois - Theater Director.)
- RF
QUOTE (Raphie Frank+Aug 25 2009, 12:46 PM)
Umm... Gothamist.com, named Top City Blog by Forbes Magazine, The Evening Standard (of London... over hald a page in the "Going Out" section...), Time Out London...
Damn, things are going over your head quickly. The salient point was that "Paperclips-Monthly" is not an academic journal.
And again you avoid any mention of the mathematics in my posts? Come on, Raphie Frank, let's see some of your "number researcher" skills...
Damn, things are going over your head quickly. The salient point was that "Paperclips-Monthly" is not an academic journal.
And again you avoid any mention of the mathematics in my posts? Come on, Raphie Frank, let's see some of your "number researcher" skills...
My eyes are watering. I've just googled "Raphie Frank", and this is what I got:
More at (http://artivist.blogspot.com/). Please, please tell me this isn't you?
QUOTE
We ourselves are the space and the connection between things. As such, welcome to the 5th dimension aka the eternal "now" and the psychosocial eqivalent to e=mc^2 and perhaps far more:
Z= PI2
Zightgeist = Power * speed of Idea2
Network, system and string theory are the missing links. Also, for what it's worth, my derivations for Z=PI2 predict that light can travel backwards as well as that 1=2 (aka the circular paradox wherein the formaula for the Area can reduce to 1 or 2). But first you've got to accept that 0 actually = -1 and is the opposite side of +1. In physics 101 it's called potential energy.
Z= PI2
Zightgeist = Power * speed of Idea2
Network, system and string theory are the missing links. Also, for what it's worth, my derivations for Z=PI2 predict that light can travel backwards as well as that 1=2 (aka the circular paradox wherein the formaula for the Area can reduce to 1 or 2). But first you've got to accept that 0 actually = -1 and is the opposite side of +1. In physics 101 it's called potential energy.
More at (http://artivist.blogspot.com/). Please, please tell me this isn't you?
QUOTE (Euler+Aug 25 2009, 12:37 PM)
But now you've changed the equation. It is easy to prove that the only solutions to your new equation are n=0,3,5.
Wonderful, Euler. Good to hear. And how does one prove this, especially given that 1 is also a solution (an oversight on your part I am sure... (1^3 - 1)/4 = 0 = 2^1 - 2)?
Please do consider this, however, a special case of the original question.
I removed the "Ceiling" condition only to make the question more accessible and less "controversial."
Wonderful, Euler. Good to hear. And how does one prove this, especially given that 1 is also a solution (an oversight on your part I am sure... (1^3 - 1)/4 = 0 = 2^1 - 2)?
Please do consider this, however, a special case of the original question.
I removed the "Ceiling" condition only to make the question more accessible and less "controversial."
QUOTE (Raphie Frank+Aug 25 2009, 12:54 PM)
Wonderful, Euler. Good to hear. And how does one prove this, especially given that 1 is also a solution (an oversight on your part I am sure... (1^3 - 1)/4 = 0 = 2^1 - 2)?
Surely you can prove this? High school students could prove this, so I expect a number researcher to be able to do so. I, merely someone who writes papers, thinks the proof is trivial. Surely you, some who is written about, can at least manage an attempt? Come on...
Surely you can prove this? High school students could prove this, so I expect a number researcher to be able to do so. I, merely someone who writes papers, thinks the proof is trivial. Surely you, some who is written about, can at least manage an attempt? Come on...
QUOTE (Raphie Frank+Aug 25 2009, 12:54 PM)
I removed the "Ceiling" condition only to make the question more accessible and less "controversial."
But then this is a completely different question. I have told you how you can justify the removal of the ceiling function, by considering two special cases mod8. I've even written it all out for you. Why can't you figure out what to do?
Let's see the number researcher at work. By the way, it might be beneficial if you don't rely on your claim
But then this is a completely different question. I have told you how you can justify the removal of the ceiling function, by considering two special cases mod8. I've even written it all out for you. Why can't you figure out what to do?
Let's see the number researcher at work. By the way, it might be beneficial if you don't rely on your claim
QUOTE (Raphie Frank+)
my derivations for Z=PI2 predict that light can travel backwards as well as that 1=2 (aka the circular paradox wherein the formaula for the Area can reduce to 1 or 2). But first you've got to accept that 0 actually = -1 and is the opposite side of +1
Deep, though it is.
Deep, though it is.
QUOTE (Euler+Aug 25 2009, 12:53 PM)
My eyes are watering. I've just googled "Raphie Frank", and this is what I got:
More at (http://artivist.blogspot.com/). Please, please tell me this isn't you?
That is absolutely me, Euler, although what you posted is the purely philosophical case and ought not to be construed in a strict mathematical light.
Z = P*I^2 = Power * the Speed of Information^2
... is the proper (current) form of my anthropomorphized equivalent to e=mc^2
It is to be taken, for now, as a loose metaphor implying relationships that could be empirically validated via social research. For instance, as information "flow" increases, overall human productivity should also increase. Similarly, extreme intelligence and psychological disorders related to coping should also increase. Most importantly, the formulation suggests that the properly directed study of Psyche and Society could yield results of import to the Hard Sciences... (and vice versa, with qualifications...)
Furthermore, the metaphor is relative in nature, applying only to an idealized closed system relative to another closed system.
For instance, where I = 0, then Z must necessarily = 0 (i.e "Silence = Death")
P is associated with e and by extension, Power laws, while I is associated with the Golden Mean (phi) and Information transfer, a notion far less absurd now than a year ago, given a recent study by a team of researchers, none of whom attended "The Paperclip Institute," although one does hold court at The Max Planck Institute, an organization I am sure our moderator, at the very least, is familiar with.
- RF
More at (http://artivist.blogspot.com/). Please, please tell me this isn't you?
That is absolutely me, Euler, although what you posted is the purely philosophical case and ought not to be construed in a strict mathematical light.
Z = P*I^2 = Power * the Speed of Information^2
... is the proper (current) form of my anthropomorphized equivalent to e=mc^2
It is to be taken, for now, as a loose metaphor implying relationships that could be empirically validated via social research. For instance, as information "flow" increases, overall human productivity should also increase. Similarly, extreme intelligence and psychological disorders related to coping should also increase. Most importantly, the formulation suggests that the properly directed study of Psyche and Society could yield results of import to the Hard Sciences... (and vice versa, with qualifications...)
Furthermore, the metaphor is relative in nature, applying only to an idealized closed system relative to another closed system.
For instance, where I = 0, then Z must necessarily = 0 (i.e "Silence = Death")
P is associated with e and by extension, Power laws, while I is associated with the Golden Mean (phi) and Information transfer, a notion far less absurd now than a year ago, given a recent study by a team of researchers, none of whom attended "The Paperclip Institute," although one does hold court at The Max Planck Institute, an organization I am sure our moderator, at the very least, is familiar with.
- RF
QUOTE (Raphie Frank+Aug 25 2009, 01:06 PM)
That is absolutely me, Euler
Ha - you are an idiot, and your daft enough to document it. Well played.
Ha - you are an idiot, and your daft enough to document it. Well played.
QUOTE (Euler+Aug 25 2009, 01:07 PM)
Ha - you are an idiot, and your daft enough to document it. Well played.
Of course I documented it, Euler. As early as January 2006, in fact, long before I had read the work of, for instance, either Albert Laszlo Barabasi, Mark Granovetter, E.O. Wilson or others.
Just last year, Princeton Professor of Sociology Matt Salganik was a session leader at a conference at the Santa Fe Institute...
Is There a Physics of Society?
January 10-12, 2008, Santa Fe NM
http://www.santafe.edu/events/workshops/in...ety%3F_-_Agenda
To be "right" and to be on "the right track" are two very different things. I believe myself on "the right track" and do not wish to be told by others, later, that I am only at that time ascribing to certain ideas I may ascribe to at that time because I am "name-dropping" "hot" concepts or topics.
The question, Euler, is not IF, but to what degree, one can apply Physics to the EFFECTIVE study of human interaction on local or global level. The general notion, not in regards my specific formulation, but in regards to the overarching question of "Can the study of Physics be profitable applied to the study of human interaction" is not speculative, but has already been shown to be the case in the affirmative via empirical research.
- RF
Of course I documented it, Euler. As early as January 2006, in fact, long before I had read the work of, for instance, either Albert Laszlo Barabasi, Mark Granovetter, E.O. Wilson or others.
Just last year, Princeton Professor of Sociology Matt Salganik was a session leader at a conference at the Santa Fe Institute...
Is There a Physics of Society?
January 10-12, 2008, Santa Fe NM
http://www.santafe.edu/events/workshops/in...ety%3F_-_Agenda
To be "right" and to be on "the right track" are two very different things. I believe myself on "the right track" and do not wish to be told by others, later, that I am only at that time ascribing to certain ideas I may ascribe to at that time because I am "name-dropping" "hot" concepts or topics.
The question, Euler, is not IF, but to what degree, one can apply Physics to the EFFECTIVE study of human interaction on local or global level. The general notion, not in regards my specific formulation, but in regards to the overarching question of "Can the study of Physics be profitable applied to the study of human interaction" is not speculative, but has already been shown to be the case in the affirmative via empirical research.
- RF
QUOTE (Raphie Frank+Aug 25 2009, 01:35 PM)
Of course I documented it, Euler. As early as January 2006, in fact, long before I had read the work of, for instance, either Albert Laszlo Barabasi, Mark Granovetter, E.O. Wilson or others.
So these guys are writing things like:
So these guys are writing things like:
QUOTE (Raphie Frank+)
my derivations for Z=PI2 predict that light can travel backwards as well as that 1=2 (aka the circular paradox wherein the formaula for the Area can reduce to 1 or 2). But first you've got to accept that 0 actually = -1 and is the opposite side of +1
So let's just summarise:
So let's just summarise:
- You've predicted light can travel backwards. That's amazing - I thought it only went "forwards"!
- You derived that 1=2. Wow, we're really onto the good stuff now
- You need people to accept that -1=0. Well why on earth wouldn't they!
QUOTE (Euler+Aug 25 2009, 01:42 PM)
So let's just summarise:[LIST]
[*]You've predicted light can travel backwards. That's amazing - I thought it only went "forwards"!
OFF-TOPIC
I made this "prediction" around the same time the following article came out...
===========================================
May 11, 2006 Contact: Jonathan Sherwood
jonathan.sherwood@rochester.edu
585.273.4726
Light's Most Exotic Trick Yet: So Fast it Goes ... Backwards?
In the past few years, scientists have found ways to make light go both faster and slower than its usual speed limit, but now researchers at the University of Rochester have published a paper today in Science on how they've gone one step further: pushing light into reverse. As if to defy common sense, the backward-moving pulse of light travels faster than light.
Confused? You're not alone.
MORE: http://www.rochester.edu/news/show.php?id=2544
===========================================
I do not necessarily subscribe or not subscribe to that point of view now, and do not remember if it was before or after the article was published that I made the "prediction," just that the article confirmed the ideas I was already toying with at the time and was making known (and documenting), for instance, to close friends and family members. Naturally, this article excited me.
My logic at the time has long since faded, as my "mathematical" endeavors at that time consisted primarily of drawings and quite obsure notations that upon review even I do not now understand. Such scribblings came in the aftermath of November 28th, 2005 when I experienced a phenomeneon the general public calls a "light bulb" moment.
For myself I call such a moment, POETICALLY, "fission with vision," and in all honesty, it was more than a bit overwhelming at the time, ESPECIALLY because I understood, quite well, how "ridiculous" such ideas (e.g. Z = P*I^2) were relative to common sense wisdom
As such, I might add, I was quite relieved to find that Richard Dawkins, in loose manner, beat me to the punch in 1998 when he published "Unweaving the Rainbow," , a fact I only "discovered" just last year. In that book he proposes a concept he terms "self-feeding co-evolution."
Although disappointed not to be "first," I was also quite relieved to know that a reputable figure, any reputable figure, had proposed similar ideas in relation to human dynamic. Mark Granovetter, a physicist, also proposed a similar "nuclear chain reaction" mechanism at work in relation to social phenomeneon, in his book "Nexus" Small Worlds and the Groundbreaking Theory of Networks" which I finally came across in late 2008 or early 2009.
So...
What can I say of myself, Euler? That I believe in Physics and Science? That I do not believe in God?
Both are true.
Press this, if you wish, but this is extremely off-topic... I wanted, however, to give you and others, an honest response, for I recognize how controversial such ideas may be.
Such ideas imply (possibly), for instance, that the technological progression of Humankind (especially, I might add, our units of measurement, which can be thought of in similar manner to a Nation's "Gross Domestic Product," which "flatlines" complexity into a lump sum...), is no less subject to natural law than bees in a honeycomb or molecules of water in a pot on a heating stove approaching the boiling point.
Extremely controversial, I know, and because of this I have gone out of my way not to press such notions upon this forum.
Best,
Raphie
[*]You've predicted light can travel backwards. That's amazing - I thought it only went "forwards"!
OFF-TOPIC
I made this "prediction" around the same time the following article came out...
===========================================
May 11, 2006 Contact: Jonathan Sherwood
jonathan.sherwood@rochester.edu
585.273.4726
Light's Most Exotic Trick Yet: So Fast it Goes ... Backwards?
In the past few years, scientists have found ways to make light go both faster and slower than its usual speed limit, but now researchers at the University of Rochester have published a paper today in Science on how they've gone one step further: pushing light into reverse. As if to defy common sense, the backward-moving pulse of light travels faster than light.
Confused? You're not alone.
MORE: http://www.rochester.edu/news/show.php?id=2544
===========================================
I do not necessarily subscribe or not subscribe to that point of view now, and do not remember if it was before or after the article was published that I made the "prediction," just that the article confirmed the ideas I was already toying with at the time and was making known (and documenting), for instance, to close friends and family members. Naturally, this article excited me.
My logic at the time has long since faded, as my "mathematical" endeavors at that time consisted primarily of drawings and quite obsure notations that upon review even I do not now understand. Such scribblings came in the aftermath of November 28th, 2005 when I experienced a phenomeneon the general public calls a "light bulb" moment.
For myself I call such a moment, POETICALLY, "fission with vision," and in all honesty, it was more than a bit overwhelming at the time, ESPECIALLY because I understood, quite well, how "ridiculous" such ideas (e.g. Z = P*I^2) were relative to common sense wisdom
As such, I might add, I was quite relieved to find that Richard Dawkins, in loose manner, beat me to the punch in 1998 when he published "Unweaving the Rainbow," , a fact I only "discovered" just last year. In that book he proposes a concept he terms "self-feeding co-evolution."
Although disappointed not to be "first," I was also quite relieved to know that a reputable figure, any reputable figure, had proposed similar ideas in relation to human dynamic. Mark Granovetter, a physicist, also proposed a similar "nuclear chain reaction" mechanism at work in relation to social phenomeneon, in his book "Nexus" Small Worlds and the Groundbreaking Theory of Networks" which I finally came across in late 2008 or early 2009.
So...
What can I say of myself, Euler? That I believe in Physics and Science? That I do not believe in God?
Both are true.
Press this, if you wish, but this is extremely off-topic... I wanted, however, to give you and others, an honest response, for I recognize how controversial such ideas may be.
Such ideas imply (possibly), for instance, that the technological progression of Humankind (especially, I might add, our units of measurement, which can be thought of in similar manner to a Nation's "Gross Domestic Product," which "flatlines" complexity into a lump sum...), is no less subject to natural law than bees in a honeycomb or molecules of water in a pot on a heating stove approaching the boiling point.
Extremely controversial, I know, and because of this I have gone out of my way not to press such notions upon this forum.
Best,
Raphie
QUOTE (prometheus+Aug 25 2009, 12:58 PM)
While we're at it, I've spoken to Hawking. He didn't speak to me though, I was most insulted!
Well I beat him in an arm wrestling contest!
Well I beat him in an arm wrestling contest!
QUOTE (Raphie Frank+Aug 25 2009, 12:58 PM)
So, uh.. wait. To be "published" is worthy of bragging rights in mathematics, but not other fields? To the point that no less than 3 posters commented upon this in a spirit of ridicule within the span of 1/2 hour?
When any of us say 'published' we mean that work we've done, which noone else has done before, has been put into an academically relevant and reputable journal, having pass peer review, so that others might benefit from it. That's entirely different from writing an article on your views about welfare reform or interviewing anyone. Journalists, after all, are not academically gifted people, though some are it wasn't the reason they got their job nor does their job imply they are. To be 'published' in the sense we use the word is to mean your original research has been deemed to be informative and useful by other academics. Writing an article on the football game you watched on the weekend is utterly different. Anyone can have opinions, informed or not, stupid or not, and anyone can conduct an interview. Yes, some people do it better than others but the fact is its not equivalent to making a valid contribution to the field of human knowledge.
QUOTE (Raphie Frank+Aug 25 2009, 12:58 PM)
although what you posted is the purely philosophical case and ought not to be construed in a strict mathematical light.
Except it is quite clear you want it to be viewed in a mathematical light. IF you didn't you'd not be using mathematical terminology. It's entirely possible to do amazing philosophy without making it look like you're trying to do mathematics.
It's very clear you want people to believe you're mathematically inclined and knowledgable but when anyone scratches below the surface it's clear you are not.
It's very clear you want people to believe you're mathematically inclined and knowledgable but when anyone scratches below the surface it's clear you are not.
QUOTE (Raphie Frank+Aug 25 2009, 12:58 PM)
For instance, where I = 0, then Z must necessarily = 0 (i.e "Silence = Death")
This is just nonsense. You can make plenty of philosophical statements about death, life, the universe, existence, the soul etc without having to resort to just nonsense vapid equations.
QUOTE (Raphie Frank+Aug 25 2009, 12:58 PM)
The question, Euler, is not IF, but to what degree, one can apply Physics to the EFFECTIVE study of human interaction on local or global level. The general notion, not in regards my specific formulation, but in regards to the overarching question of "Can the study of Physics be profitable applied to the study of human interaction" is not speculative, but has already been shown to be the case in the affirmative via empirical research.
You clearly don't know any maths or physics and philosophy is completely beyond you so I wonder what exactly you are trying to do.
Simply because someone else has done something you find interesting doesn't mean that if you spew buzzword and half formed gibberish equations ad nauseum that somehow you're doing science too. This point is illustrated in the post directly above this on. You bounce from Dawkins and evolution through to nuclear physics, pass by a vapid equation or two and end up claiming you're working along the same lines as some famous people. You think that if you throw out buzzwords you're doing the stuff Dawkins does. Do you think he got to where he is by making vapid nonsense? Would he be in the academic research position he is today if there was nothing more to his work than his books? Of course not. If you asked him to go into research level detail on anything in his work he could. You can't get past your buzzwords because you don't understand them, there is nothing below the surface.
Simply because someone else has done something you find interesting doesn't mean that if you spew buzzword and half formed gibberish equations ad nauseum that somehow you're doing science too. This point is illustrated in the post directly above this on. You bounce from Dawkins and evolution through to nuclear physics, pass by a vapid equation or two and end up claiming you're working along the same lines as some famous people. You think that if you throw out buzzwords you're doing the stuff Dawkins does. Do you think he got to where he is by making vapid nonsense? Would he be in the academic research position he is today if there was nothing more to his work than his books? Of course not. If you asked him to go into research level detail on anything in his work he could. You can't get past your buzzwords because you don't understand them, there is nothing below the surface.
Stupid question alert,
Is TT an infinite "string" of "randomly" reoccurring numbers?
Is TT an infinite "string" of "randomly" reoccurring numbers?
Maybe, it's not so stupid?
QUOTE (Meem+Aug 25 2009, 06:02 PM)
Stupid question alert,
Is TT an infinite "string" of "randomly" reoccurring numbers?
Can I assume, meem, that you are referring to Godel, Escher, Bach by Douglas Hofstadter?
If so, this thread is the wrong place for the question.
I did my part to take it off-topic and now wish to do my part to bring it back on-topic.
Best,
RF
P.S. If, in fact, you are referring to Hofstadter, by the way, I am not well enough acquainted with the "TT" system to offer a constructive answer.
Is TT an infinite "string" of "randomly" reoccurring numbers?
Can I assume, meem, that you are referring to Godel, Escher, Bach by Douglas Hofstadter?
If so, this thread is the wrong place for the question.
I did my part to take it off-topic and now wish to do my part to bring it back on-topic.
Best,
RF
P.S. If, in fact, you are referring to Hofstadter, by the way, I am not well enough acquainted with the "TT" system to offer a constructive answer.
QUOTE (Raphie Frank+Aug 25 2009, 02:25 PM)
My logic at the time has long since faded, as my "mathematical" endeavors at that time consisted primarily of drawings and quite obsure notations that upon review even I do not now understand. Such scribblings came in the aftermath of November 28th, 2005 when I experienced a phenomeneon the general public calls a "light bulb" moment.
This stuff is hilarious. But I'm also worried, because some of it doesn't make sense, and the only conclusion I can come to is that you're lying, again.
Firstly, let me explain something to you. The statement "light travels backwards" means nothing - it is empty, vacuous. Just as the statement "photons move left". It's the kind of statement you seem to like: to some it may sound profound, but in reality it is completely lacking in scientific substance. I imagine the statement was first used by the experimenters, when speaking to uninformed journalists - they wanted the uninformed majority of people to see the line "light travels backwards", and think wow! You being a member of that uninformed majority, you too think it profound. All the experimenters did was construct a scenario in which the 1 dimensional group velocity had opposite sign to the phase velocity - thus constructing a pulse which moved in the opposite direction to the propagation of the photons making the beam.
So now we've established that the phrase "light travels backwards" means nothing. But then, how would one produce a mathematical argument that predicts an empty statement? You say that you once had a derivation of your result, but (surprise, surprise) you're unable to show it off because you don't understand the notation you made up. I'd like to know how you "derived" (chuckle!) such a statement.
I can't get over what an absolute cretin you must be!
This stuff is hilarious. But I'm also worried, because some of it doesn't make sense, and the only conclusion I can come to is that you're lying, again.
Firstly, let me explain something to you. The statement "light travels backwards" means nothing - it is empty, vacuous. Just as the statement "photons move left". It's the kind of statement you seem to like: to some it may sound profound, but in reality it is completely lacking in scientific substance. I imagine the statement was first used by the experimenters, when speaking to uninformed journalists - they wanted the uninformed majority of people to see the line "light travels backwards", and think wow! You being a member of that uninformed majority, you too think it profound. All the experimenters did was construct a scenario in which the 1 dimensional group velocity had opposite sign to the phase velocity - thus constructing a pulse which moved in the opposite direction to the propagation of the photons making the beam.
So now we've established that the phrase "light travels backwards" means nothing. But then, how would one produce a mathematical argument that predicts an empty statement? You say that you once had a derivation of your result, but (surprise, surprise) you're unable to show it off because you don't understand the notation you made up. I'd like to know how you "derived" (chuckle!) such a statement.
I can't get over what an absolute cretin you must be!
Surreal...
QUOTE (Euler+Aug 26 2009, 07:55 AM)
I can't get over what an absolute cretin you must be!
CRETIN...
Too funny!
A hint for you, Euler...
In social terms, begin from the notion that a circle exists rather than trying to build one up from a set of discrete points, for if you approached from such an intellectual "angle" you would "rightly" conclude that the notion of a true circle was utterly absurd.
Never mind how wrong you would be, as your conclusion would be consistent with the set of definitions and axioms from which you had begun...
Carl Jung termed the point I am conveying [i.e begin from the assumption of the circle (i.e infinity) , a point that will be utterly lost on you...] the "final" rather than the "causal" approach.
Or... let's make it REALLY simple. How do you understand the words I am now writing you, even should you utterly and grotesquely distort them as you are wont to do through your toadish subjective lens?
Do you feel the need to "prove" each word as you go?
Have you ever heard of the term "Gestalt" young faux "Euler"?
How about "Cartesian" versus "Linear" Intelligence? You've the latter. I've the former. If you had the latter, you would understand the terms without definition...
This, on the most fundamental level, is the meaningful difference between a faux Euler and a real Raphie.... that the notion of "linear" versus "cartesian" intelligence will have as little meaning to you as an apple would have to a Flatlander.
- RF
P.S. The REAL Euler, on the other hand, and by the way, would blow us both out of the water...
CRETIN...
Too funny!
A hint for you, Euler...
In social terms, begin from the notion that a circle exists rather than trying to build one up from a set of discrete points, for if you approached from such an intellectual "angle" you would "rightly" conclude that the notion of a true circle was utterly absurd.
Never mind how wrong you would be, as your conclusion would be consistent with the set of definitions and axioms from which you had begun...
Carl Jung termed the point I am conveying [i.e begin from the assumption of the circle (i.e infinity) , a point that will be utterly lost on you...] the "final" rather than the "causal" approach.
Or... let's make it REALLY simple. How do you understand the words I am now writing you, even should you utterly and grotesquely distort them as you are wont to do through your toadish subjective lens?
Do you feel the need to "prove" each word as you go?
Have you ever heard of the term "Gestalt" young faux "Euler"?
How about "Cartesian" versus "Linear" Intelligence? You've the latter. I've the former. If you had the latter, you would understand the terms without definition...
This, on the most fundamental level, is the meaningful difference between a faux Euler and a real Raphie.... that the notion of "linear" versus "cartesian" intelligence will have as little meaning to you as an apple would have to a Flatlander.
- RF
P.S. The REAL Euler, on the other hand, and by the way, would blow us both out of the water...
Is this your way of saying you can't answer my question?
Loving the whole linear/Cartesian thing by the way. Great stuff! 1=0, circle i.e. infinity, and so on..
Loving the whole linear/Cartesian thing by the way. Great stuff! 1=0, circle i.e. infinity, and so on..
OFF-TOPIC
@ Alphanumeric
From the human intellectual perspective...
"Matter / Antimatter colliding."
That's a turn of phrase, roughy speaking, that you have used in the past.
Just a metaphor? Or perhaps an actual descriptor of human dynamic that might be mathematically described?
Consider:
If Raphie has an IQ that correlates accurately with his publicly stated high school SAT's, then must Euler and his buddy George both necessarily, by extension, be Ignoramuses, given that they have both termed him an "idiot"?
The answer is "NO."
You are not the idiot that you seem to my father, who believes me more intelligent than he (and, I might add, has read several of your posts... sorry to say, but he was not impressed, but largely because of your juvenile conduct and clearly, to him, wrong conclusions...). Nor am I the idiot you believe me to be...
Figure out a way to tell a better story, AN... rather than being so guided by your pride and blind loyalty to wrong-headed friends.
- RF
@ Alphanumeric
From the human intellectual perspective...
"Matter / Antimatter colliding."
That's a turn of phrase, roughy speaking, that you have used in the past.
Just a metaphor? Or perhaps an actual descriptor of human dynamic that might be mathematically described?
Consider:
If Raphie has an IQ that correlates accurately with his publicly stated high school SAT's, then must Euler and his buddy George both necessarily, by extension, be Ignoramuses, given that they have both termed him an "idiot"?
The answer is "NO."
You are not the idiot that you seem to my father, who believes me more intelligent than he (and, I might add, has read several of your posts... sorry to say, but he was not impressed, but largely because of your juvenile conduct and clearly, to him, wrong conclusions...). Nor am I the idiot you believe me to be...
Figure out a way to tell a better story, AN... rather than being so guided by your pride and blind loyalty to wrong-headed friends.
- RF
QUOTE (Raphie Frank+Aug 26 2009, 09:26 AM)
You are not the idiot that you seem to my father, who believes me more intelligent than he
Ahh, daddy tells you you're special? Bless your little cotton socks!
Ahh, daddy tells you you're special? Bless your little cotton socks!
QUOTE (Euler+Aug 26 2009, 09:30 AM)
Ahh, daddy tells you you're special? Bless your little cotton socks!
For the record, my father is also one of my greatest critics, Euler.
Shall I conjure up the post where Alphanumeric, aka "George," talks about his father believing him more talented in maths than he?
Ah, bless his little cotton socks!
You are a HYPOCRITE to the extreme my little anonymous (read: cowardly (IMO only)) friend.
- RF
For the record, my father is also one of my greatest critics, Euler.
Shall I conjure up the post where Alphanumeric, aka "George," talks about his father believing him more talented in maths than he?
Ah, bless his little cotton socks!
You are a HYPOCRITE to the extreme my little anonymous (read: cowardly (IMO only)) friend.
- RF
QUOTE (Raphie Frank+Aug 26 2009, 09:37 AM)
For the record, my father is also one of my greatest critics, Euler.
Shall I conjure up the post where Alphanueric talks about his father believing him more talented in maths than he?
Ah, bless his little contton socks!
But then, no one doubts AlphaNumeric's mathematical talents. I am telling you and everyone here that I think you're mathematically clueless and intellectually inept, to which you have responded "My daddy tells me I'm smart".
Oh this is fun!
Shall I conjure up the post where Alphanueric talks about his father believing him more talented in maths than he?
Ah, bless his little contton socks!
But then, no one doubts AlphaNumeric's mathematical talents. I am telling you and everyone here that I think you're mathematically clueless and intellectually inept, to which you have responded "My daddy tells me I'm smart".
Oh this is fun!
QUOTE (Euler+Aug 26 2009, 09:40 AM)
But then, no one doubts AlphaNumeric's mathematical talents. I am telling you and everyone here that I think you're mathematically clueless and intellectually inept, to which you have responded "My daddy tells me I'm smart".
Oh this is fun!
And no one but the outright delusional doubt, not my mathematical talents, but my generalized intelligence...
For the record, Rpenner has the means, and has had the means for many months now, to contact my father to, at the very least, verify my identity and general truthfulness.
- RF
P.S. "Cartesian" verseus "Linear" Intelligence, Euler. See if you can wrap your relatively 1-D intelligence around the implications of that notion...
Oh this is fun!
And no one but the outright delusional doubt, not my mathematical talents, but my generalized intelligence...
For the record, Rpenner has the means, and has had the means for many months now, to contact my father to, at the very least, verify my identity and general truthfulness.
- RF
P.S. "Cartesian" verseus "Linear" Intelligence, Euler. See if you can wrap your relatively 1-D intelligence around the implications of that notion...
QUOTE (Raphie Frank+Aug 26 2009, 09:46 AM)
And no one but the outright delusional doubt, not my mathematical talents, but my generalized intelligence...
I find it hilarious that you're surprised people don't think you're smart. Honestly, I have no reason to lie to you: I think you are an idiot. I think you are unintelligent. I think you vastly overestimate your own intellectual ability. What is more, all these things are blindingly obvious!
It is clear that this is a particularly sore point for you, and it is perhaps why you try sooo hard to be seen to be doing intellectual things. Why you insist on such flowery passages of self-indulgent, meaningless rubbish. Sad.
I find it hilarious that you're surprised people don't think you're smart. Honestly, I have no reason to lie to you: I think you are an idiot. I think you are unintelligent. I think you vastly overestimate your own intellectual ability. What is more, all these things are blindingly obvious!
It is clear that this is a particularly sore point for you, and it is perhaps why you try sooo hard to be seen to be doing intellectual things. Why you insist on such flowery passages of self-indulgent, meaningless rubbish. Sad.
QUOTE (Raphie Frank+Aug 26 2009, 09:46 AM)
For the record, Rpenner has the means, and has had the means for many months now, to contact my father to, at the very least, verify my identity and general truthfulness.
A smart person would attempt to curb the general opinion of their intellect by reducing the amount of stupid things they say and increasing the amount of intelligent things they say. Other types of people might, I don't know, ask people to contact their parents.
Come on, try harder!
A smart person would attempt to curb the general opinion of their intellect by reducing the amount of stupid things they say and increasing the amount of intelligent things they say. Other types of people might, I don't know, ask people to contact their parents.
Come on, try harder!
Over and over again you show yourself to be almost beyond help, very young Euler. Reality seems to have no place in your world... and from the basic standpoint of human decency, you are the very type of person a certain family member of mine had this to say about this past May...
"However, from what I have seen of the PhysOrg Forum, the participants frequently use abusive language and are extremely rude to each other. Very little serious intellectual interchange seems to take place. Why would you want to have anything to do with it? I would not waste my time. You have more important things to do."
(published by permission...)
Words to reflect upon...
- RF
"However, from what I have seen of the PhysOrg Forum, the participants frequently use abusive language and are extremely rude to each other. Very little serious intellectual interchange seems to take place. Why would you want to have anything to do with it? I would not waste my time. You have more important things to do."
(published by permission...)
Words to reflect upon...
- RF
And no response to the points I raise? I'd hate to think I was right on the money, and you're trying to avoid the issue...
QUOTE (Raphie Frank+Aug 26 2009, 10:06 AM)
QUOTE (Euler+Aug 26 2009, 10:11 AM)
And no response to the points I raise? I'd hate to think I was right on the money, and you're trying to avoid the issue...
Which points were those again that I ought to address, Euler?
I think the list goes something like this:
Raphie Frank is "unintelligent."
Raphie Frank is an "idiot".
Raphie Frank is a "little man."
Raphie Frank is "dishonest."
Raphie Frank is a "liar."
Raphie Frank "lacks any mathematical aptitude" (therefore, he either lied about test scores, or the tests are flawed...)
And so on... about 10* over at the least.
Where shall I begin and why on this good earth would it be worth my time to try to explain to a peacockish and proud Flatlander, that, metaphorically speaking, the world is round?
OF COURSE I SEEM STUPID TOU YOU, EULER!
I may as well be a brother Wright telling the esteemed Lord Kelvin that, "no, good sir, man CAN actually fly."
"Linear" versus "Cartesian" Intelligence. The fact that the meaning of this thought is lost on you speaks volumes.
- RF
Which points were those again that I ought to address, Euler?
I think the list goes something like this:
Raphie Frank is "unintelligent."
Raphie Frank is an "idiot".
Raphie Frank is a "little man."
Raphie Frank is "dishonest."
Raphie Frank is a "liar."
Raphie Frank "lacks any mathematical aptitude" (therefore, he either lied about test scores, or the tests are flawed...)
And so on... about 10* over at the least.
Where shall I begin and why on this good earth would it be worth my time to try to explain to a peacockish and proud Flatlander, that, metaphorically speaking, the world is round?
OF COURSE I SEEM STUPID TOU YOU, EULER!
I may as well be a brother Wright telling the esteemed Lord Kelvin that, "no, good sir, man CAN actually fly."
"Linear" versus "Cartesian" Intelligence. The fact that the meaning of this thought is lost on you speaks volumes.
- RF
QUOTE (Raphie Frank+Aug 26 2009, 10:41 AM)
I may as well be a brother Wright telling the esteemed Lord Kelvin that, "no, good sir, man CAN actually fly."
Wow - so you'd actually compare what you do, to man's achievement of flight. Again, wow.
Now, what I'd like to hear is:
1) Why have you continually avoided the mathematical points raised in this thread. Particularly, why didn't you respond to my post here regarding the removal of the ceiling function. Why did you not respond to explanation of your use of the phrase "kissing numbers" here. Why were you unable to provide the 2-second proof to the result mentioned here.
2) Why were you unable to back up your story, by responding to my post here. If you just want to admit you were talking rubbish, again, then that's fine - we're used to it.
3) You failed to backup the statements (which you've confirmed you are responsible for) -1=0 and 1=2, as I pointed out here.
4) Why were you unable to respond to this, which outlines your actions and the natural conclusions which arise from them.
There are umpteen more of these, in this thread alone. Your general tactic on this forum, is to write spurious nonsense, then avoid addressing any point that is made which indicates you've been talking utter tripe. Let's not forget when you made a fool of yourself, again, when trying to tell everyone that you understood what a Lie group was.
I look forward to your detailed responses...
Wow - so you'd actually compare what you do, to man's achievement of flight. Again, wow.
Now, what I'd like to hear is:
1) Why have you continually avoided the mathematical points raised in this thread. Particularly, why didn't you respond to my post here regarding the removal of the ceiling function. Why did you not respond to explanation of your use of the phrase "kissing numbers" here. Why were you unable to provide the 2-second proof to the result mentioned here.
2) Why were you unable to back up your story, by responding to my post here. If you just want to admit you were talking rubbish, again, then that's fine - we're used to it.
3) You failed to backup the statements (which you've confirmed you are responsible for) -1=0 and 1=2, as I pointed out here.
4) Why were you unable to respond to this, which outlines your actions and the natural conclusions which arise from them.
There are umpteen more of these, in this thread alone. Your general tactic on this forum, is to write spurious nonsense, then avoid addressing any point that is made which indicates you've been talking utter tripe. Let's not forget when you made a fool of yourself, again, when trying to tell everyone that you understood what a Lie group was.
I look forward to your detailed responses...
QUOTE (Euler+Aug 26 2009, 11:02 AM)
Wow - so you'd actually compare what you do, to man's achievement of flight. Again, wow.
Now, what I'd like to hear is:
1) Why have you continually avoided the mathematical points raised in this thread. Particularly, why didn't you respond to my post here regarding the removal of the ceiling function. Why did you not respond to explanation of your use of the phrase "kissing numbers" here. Why were you unable to provide the 2-second proof to the result mentioned here.
2) Why were you unable to back up your story, by responding to my post here. If you just want to admit you were talking rubbish, again, then that's fine - we're used to it.
3) You failed to backup the statements (which you've confirmed you are responsible for) -1=0 and 1=2, as I pointed out here.
4) Why were you unable to respond to this, which outlines your actions and the natural conclusions which arise from them.
There are umpteen more of these, in this thread alone. Your general tactic on this forum, is to write spurious nonsense, then avoid addressing any point that is made which indicates you've been talking utter tripe. Let's not forget when you made a fool of yourself, again, when trying to tell everyone that you understood what a Lie group was.
I look forward to your detailed responses...
Matter vs. Antimatter... Which be which is, of course, a relative notion...
Suffice it to state that, a) past statements I made in a non mathematical forum context, and explicitly stated upon this thread ought not to be taken literally, but poetically, are being taken literally; and b ) that I already explained quite clearly in layman's terms the basic principle of a Lie Group. (For a clear explanation by an expert, see "Why Beauty is Truth" p. 163).
- RF
P.S. Just curious, Euler, but you are under 5 feet 9 inches, right? I only suggest this to be so as you manifest many Adlerian-type symptoms of psychic compensenation for self-perceived physical defect... In plain, but oversimplified, English, you (IMO) are a classic case of the Napoleon complex. The classic projective psychology tip off, of course, being that you called me a "little man." A bit of projection going on there, Euler?
Now, what I'd like to hear is:
1) Why have you continually avoided the mathematical points raised in this thread. Particularly, why didn't you respond to my post here regarding the removal of the ceiling function. Why did you not respond to explanation of your use of the phrase "kissing numbers" here. Why were you unable to provide the 2-second proof to the result mentioned here.
2) Why were you unable to back up your story, by responding to my post here. If you just want to admit you were talking rubbish, again, then that's fine - we're used to it.
3) You failed to backup the statements (which you've confirmed you are responsible for) -1=0 and 1=2, as I pointed out here.
4) Why were you unable to respond to this, which outlines your actions and the natural conclusions which arise from them.
There are umpteen more of these, in this thread alone. Your general tactic on this forum, is to write spurious nonsense, then avoid addressing any point that is made which indicates you've been talking utter tripe. Let's not forget when you made a fool of yourself, again, when trying to tell everyone that you understood what a Lie group was.
I look forward to your detailed responses...
Matter vs. Antimatter... Which be which is, of course, a relative notion...
Suffice it to state that, a) past statements I made in a non mathematical forum context, and explicitly stated upon this thread ought not to be taken literally, but poetically, are being taken literally; and b ) that I already explained quite clearly in layman's terms the basic principle of a Lie Group. (For a clear explanation by an expert, see "Why Beauty is Truth" p. 163).
- RF
P.S. Just curious, Euler, but you are under 5 feet 9 inches, right? I only suggest this to be so as you manifest many Adlerian-type symptoms of psychic compensenation for self-perceived physical defect... In plain, but oversimplified, English, you (IMO) are a classic case of the Napoleon complex. The classic projective psychology tip off, of course, being that you called me a "little man." A bit of projection going on there, Euler?
That's strange. It's as if, once again, you're unable to address some very clear questions, and have responded with some drivel. Are you trying to help illustrate the point I've raised?
QUOTE (Euler+Aug 26 2009, 11:32 AM)
That's strange. It's as if, once again, you're unable to address some very clear questions, and have responded with some drivel. Are you trying to help illustrate the point I've raised?
I just glanced at the links , Euler, and your points are not at all worth addressing. They are silly and quite plainly DUMB, especially given that you made a big to-do of publicly noting that you believed the proof exceptionally difficult before flipping 180, but only, of course, after I removed the ceiling condition...
What a petty small-minded hypocrite you are.
- RF
I just glanced at the links , Euler, and your points are not at all worth addressing. They are silly and quite plainly DUMB, especially given that you made a big to-do of publicly noting that you believed the proof exceptionally difficult before flipping 180, but only, of course, after I removed the ceiling condition...
What a petty small-minded hypocrite you are.
- RF
QUOTE (Raphie Frank+Aug 26 2009, 11:45 AM)
I just glanced at the links , Euler, and your points are not at all worth addressing. They are silly and quite plainly DUMB, especially given that you made a big to-do of publicly noting that you believed the proof exceptionally difficult before flipping 180, but only, of course, after I removed the ceiling condition...
So just to clarify - you can't respond to any of them? Why is this a surprise...
[And please read my posts a little more carefully: the easy problem is to classify solutions of (n^3-n)=4(2^n-2), which, as I mentioned, is trivial]
So just to clarify - you can't respond to any of them? Why is this a surprise...
[And please read my posts a little more carefully: the easy problem is to classify solutions of (n^3-n)=4(2^n-2), which, as I mentioned, is trivial]
QUOTE (Raphie Frank+Aug 26 2009, 10:26 AM)
Figure out a way to tell a better story, AN... rather than being so guided by your pride and blind loyalty to wrong-headed friends.
Nothing to do with my loyalty to Euler. If he was being an *** and saying stuff I disagree with I'd tell him. My comments about you I stand by. You do want people to take your posts in a mathematical light, to view you as someone whose 'working at the coal face' of number theory, independently and perhaps battling against the mainstream.
You have found us less than receptive to your mathematical musings and the fact you try to fall back onto such things as how you've interviewed people or rambling about 'Cartesian intelligence' (which only has 151 hits on Google....) rather than putting your maths where your mouth is. Euler and I don't deny your test scores, we deny their relevancy (and I have explained why previously). We don't deny your interviewing history, we deny its relevancy.
The fastest, simplest way to get us to change what we say is to demonstrate we're wrong. For instance, if Euler and I didn't know one another and one of us were a new member here I'm sure we'd view the comments of the other as a touch arrogant (a trait I don't deny in our posting styles) and might say "What makes you say that about [technical subject being discussed]? How'd you go about proving that?" and the important, critical difference between the reply one of us would provide and the kind of replies you provide is that it'd be on topic, directly relevant and factually accurate. You deliberately go off on tangents whenever we press home a point about your mathematical musings. No one gives a whoot about your 'linear intelligence vs Cartesian intelligence'. The label you apply to how you view our intelligence/work/knowledge compared to yours or others is immaterial. You can call our intelligence 'Crack head space monkey intelligence' for all I care, the simple fact of the matter will be that when push comes to shove we can come up with the goods, such as Euler basically giving you the answer to your Ceiling[(x^3-x)/4] = 2^n - 2 query. You went round and round in circles, restating the same thing in half a dozen different ways but got no closer to the solution than when you first asked the question.
You post threads like this wanting people to say to you "Wow, you're a real whiz at this!". It's obvious from the lack of other replies here that few to none persons wish to discuss the actual mathematics with you so you can't be posting your mathematical musings for the purposes of discussion (that only works for the first few posts). You don't wish to engage in conversation with anyone who says you're going about things incorrectly, thus you aren't interested in learning. So we're left with you wanting a slap on the back from someone you've swindled into believing you grasp this kind of stuff. If I'm wrong about this please explain why you post your mathematical musings.
Euler and I don't bother to post our work because it'd not generate much discussion other than "I don't understand it, can you explain it in simpler terms?" or nutty cranks throwing in their nonsense. It would only serve to be a "Hey, look at the tough work I do, are I the best thing since sliced bread!". In real life I actually avoid discussing physics with anyone outside the office because I don't want to come across as trying to say "Yeah, my work is really hard, aren't I awesome!", I just say "I'm a student..... still."
Of course if that isn't how you want to come across as then you might want to seriously rethink why you're posting threads like this.
Nothing to do with my loyalty to Euler. If he was being an *** and saying stuff I disagree with I'd tell him. My comments about you I stand by. You do want people to take your posts in a mathematical light, to view you as someone whose 'working at the coal face' of number theory, independently and perhaps battling against the mainstream.
You have found us less than receptive to your mathematical musings and the fact you try to fall back onto such things as how you've interviewed people or rambling about 'Cartesian intelligence' (which only has 151 hits on Google....) rather than putting your maths where your mouth is. Euler and I don't deny your test scores, we deny their relevancy (and I have explained why previously). We don't deny your interviewing history, we deny its relevancy.
The fastest, simplest way to get us to change what we say is to demonstrate we're wrong. For instance, if Euler and I didn't know one another and one of us were a new member here I'm sure we'd view the comments of the other as a touch arrogant (a trait I don't deny in our posting styles) and might say "What makes you say that about [technical subject being discussed]? How'd you go about proving that?" and the important, critical difference between the reply one of us would provide and the kind of replies you provide is that it'd be on topic, directly relevant and factually accurate. You deliberately go off on tangents whenever we press home a point about your mathematical musings. No one gives a whoot about your 'linear intelligence vs Cartesian intelligence'. The label you apply to how you view our intelligence/work/knowledge compared to yours or others is immaterial. You can call our intelligence 'Crack head space monkey intelligence' for all I care, the simple fact of the matter will be that when push comes to shove we can come up with the goods, such as Euler basically giving you the answer to your Ceiling[(x^3-x)/4] = 2^n - 2 query. You went round and round in circles, restating the same thing in half a dozen different ways but got no closer to the solution than when you first asked the question.
You post threads like this wanting people to say to you "Wow, you're a real whiz at this!". It's obvious from the lack of other replies here that few to none persons wish to discuss the actual mathematics with you so you can't be posting your mathematical musings for the purposes of discussion (that only works for the first few posts). You don't wish to engage in conversation with anyone who says you're going about things incorrectly, thus you aren't interested in learning. So we're left with you wanting a slap on the back from someone you've swindled into believing you grasp this kind of stuff. If I'm wrong about this please explain why you post your mathematical musings.
Euler and I don't bother to post our work because it'd not generate much discussion other than "I don't understand it, can you explain it in simpler terms?" or nutty cranks throwing in their nonsense. It would only serve to be a "Hey, look at the tough work I do, are I the best thing since sliced bread!". In real life I actually avoid discussing physics with anyone outside the office because I don't want to come across as trying to say "Yeah, my work is really hard, aren't I awesome!", I just say "I'm a student..... still."
Of course if that isn't how you want to come across as then you might want to seriously rethink why you're posting threads like this.
Again, just another stupid Meem question, but maths which ever concepts practiced don't technically exist ... like time correct? Both are just concepts "invented by the mind." "The philosophy of numbers." I can have one (or any other number) of a thing, but never a one itself, because one doesn't technically exist. 1 equals one for a lot of people, but on planet x the symbol for one is actually 182734hgh345, and they have figured out many things with this ... overly complicated system, but if you ask them to show you "one" they hold up 182734hgh345 fingers! How can you really "prove" some-one wrong or right if there are different working concepts being practiced?
The first part is somewhat serious, the second not so much. TT was my attempt to first create the universe, then slice the apple "pie."
The first part is somewhat serious, the second not so much. TT was my attempt to first create the universe, then slice the apple "pie."
QUOTE (Meem+Aug 27 2009, 09:19 AM)
Again, just another stupid Meem question, but maths which ever concepts practiced don't technically exist ... like time correct? Both are just concepts "invented by the mind." "The philosophy of numbers." I can have one (or any other number) of a thing, but never a one itself, because one doesn't technically exist.
Math operates with the same given principles as the universe. 1+1 always equals 2, so on and so forth. All of physics are based on math. If you've ever studied vectors, force/energy equations, you should have seen this. All sciences are invariably based on math. It is the backbone of all knowledge. Math is a system; a set of procedures. It is a way of knowing. Math is based on nature. Nature follows mathematical rules and procedures all the time.
Time is not an invention of the mind any more than it is an invention of rocks, trees, or anything with a memory.
Math operates with the same given principles as the universe. 1+1 always equals 2, so on and so forth. All of physics are based on math. If you've ever studied vectors, force/energy equations, you should have seen this. All sciences are invariably based on math. It is the backbone of all knowledge. Math is a system; a set of procedures. It is a way of knowing. Math is based on nature. Nature follows mathematical rules and procedures all the time.
Time is not an invention of the mind any more than it is an invention of rocks, trees, or anything with a memory.
I disagree.
Math works to its own stated logical principles.
1 + 1 = 2 if 1, + , and 2 have the usual definitions.
The universe works to what appears to be consistent principles which are reconcilable with logic. And, at the human scale, the universe appears to act smoothly, which means that everything can be approximately modeled by linear or quadratic terms as long as your inputs stay close to zero.
But our first mathematically-inspired guess, Newton's is wrong in the details when speeds are fast*, and wrong altogether when dealing with the constituents of light and matter.
*[size=1]0.5 c + 0.5 c ≠ c when adding velocities (0.8 c is the actual result), but
0.5 + 0.5 = 1 when adding rapidity. (Rapidity 0.5 corresponds to (0.46211715726+)c and rapidity 1 corresponds to (0.76159415596-)c.)[/size=1]
So math still works, but not any old math. And when you trust common intuitions instead of actually looking at experimental data, you always come back to linear, quadratic and Newtonian.
So while the universe looks mathy, that's a hypothesis to be tested by science, not assumed (well at least with the leading edge of physics). But physics looks mathy always, because to make predictions from theory you have to have a rigorous logical scheme on which to make predictions and the language and operating environment of such logical system is math.
Math works to its own stated logical principles.
1 + 1 = 2 if 1, + , and 2 have the usual definitions.
The universe works to what appears to be consistent principles which are reconcilable with logic. And, at the human scale, the universe appears to act smoothly, which means that everything can be approximately modeled by linear or quadratic terms as long as your inputs stay close to zero.
But our first mathematically-inspired guess, Newton's is wrong in the details when speeds are fast*, and wrong altogether when dealing with the constituents of light and matter.
*[size=1]0.5 c + 0.5 c ≠ c when adding velocities (0.8 c is the actual result), but
0.5 + 0.5 = 1 when adding rapidity. (Rapidity 0.5 corresponds to (0.46211715726+)c and rapidity 1 corresponds to (0.76159415596-)c.)[/size=1]
So math still works, but not any old math. And when you trust common intuitions instead of actually looking at experimental data, you always come back to linear, quadratic and Newtonian.
So while the universe looks mathy, that's a hypothesis to be tested by science, not assumed (well at least with the leading edge of physics). But physics looks mathy always, because to make predictions from theory you have to have a rigorous logical scheme on which to make predictions and the language and operating environment of such logical system is math.
QUOTE (Euler+Aug 25 2009, 11:34 AM)
Give a child the equation x^n+y^n=z^n, with n>2, and leave them with it for an hour. They'll guess it can't be solved - wow, huge achievement. Can they prove it? No.
For the general advancement of knowledge...
Euler is referring to Fermat's Last Theorem.
http://en.wikipedia.org/wiki/Fermat's_Last_Theorem
- RF
For the general advancement of knowledge...
Euler is referring to Fermat's Last Theorem.
http://en.wikipedia.org/wiki/Fermat's_Last_Theorem
- RF
So, that's no response to the points I've raised, and no response to the long post AlphaNumeric made?
If what's been said isn't close to the mark, then please let us all know.
If what's been said isn't close to the mark, then please let us all know.
QUOTE (Euler+Aug 31 2009, 07:35 PM)
So, that's no response to the points I've raised, and no response to the long post AlphaNumeric made?
I am under no obligation to respond to every critique of me upon this forum, by you, by Alphanumeric or anyone else. If I were, one could simply critique me or anyone else endlessly, and then demand satisfaction. All the better if one makes unreasonable demands, thus ensuring that the "attacked" party, not only will have allowed the "opponent" to have framed the issue, whatever that issue may be, but will also be too busy countering wild goose chase assertions to offer one's own constructive voice...
I am certain you are familar with one of the classic legal strategies, especially for those without a reasonable case, Euler? One simply files petty, time consuming challenges ad nauseum to every and any statement or motion by the "enemy."
And then one challenges every response to those challenges...
Note the quotes, Euler. "Enemy" "Attack" "Opponent." Etc.
This is the modality of thought in which you (seem to) approach engagement with others.
You win only if I lose.
I win only if others also win.
This is one of the fundamental differences between us.
- RF
I am under no obligation to respond to every critique of me upon this forum, by you, by Alphanumeric or anyone else. If I were, one could simply critique me or anyone else endlessly, and then demand satisfaction. All the better if one makes unreasonable demands, thus ensuring that the "attacked" party, not only will have allowed the "opponent" to have framed the issue, whatever that issue may be, but will also be too busy countering wild goose chase assertions to offer one's own constructive voice...
I am certain you are familar with one of the classic legal strategies, especially for those without a reasonable case, Euler? One simply files petty, time consuming challenges ad nauseum to every and any statement or motion by the "enemy."
And then one challenges every response to those challenges...
Note the quotes, Euler. "Enemy" "Attack" "Opponent." Etc.
This is the modality of thought in which you (seem to) approach engagement with others.
You win only if I lose.
I win only if others also win.
This is one of the fundamental differences between us.
- RF
So again, that's a "no, I'm unable to counter your points". If you could be a little more succinct next time - no one likes to see you squirm.
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