QUOTE (Raphie Frank+Sep 26 2008, 05:18 PM)
No two numbers with the same number of divisors can be factors of one another. This is very basic, since the greater number contains at the very least all the factors of the lower number plus itself, thus the divisor number d(n) of any number that divides a greater number must be at most (n -1).
e.g. 2 has 2 divisors, 4 has 3 divisors, 8 has 4 divisors, etc.
A ) Is there a formal name for this principle?
B ) "Relatively prime" numbers are two numbers that share no common factors. In similar, but different vein, is there a name for the set of all numbers sharing the same number of divisors, none of which can be factors of or multiples of any other number in that set?
Any constructive responses would be most appreciated.
Thank you in advance.
Best,
Raphie
A) I doubt it, because it seems trivial.
B) I doubt it, because it seems pointless.
B) I doubt it, because it seems pointless.
Given that one can map the primes to each set of integers sharing a common divisor number, my response bm1957 would be that if primes are pointless, then so too any set of numbers that can map on to them.
Since numbers take up no space, however, any set of numbers, prime or otherwise, can certainly be thought of as "pointless."
But I'm pretty sure that's not the way you meant the term :-)
Best,
Raphie
Since numbers take up no space, however, any set of numbers, prime or otherwise, can certainly be thought of as "pointless."
But I'm pretty sure that's not the way you meant the term :-)
Best,
Raphie
Likewise, any attempt at rational discussion here is pointless.
That is a shame, too, 'cause this place used to be fun...
That is a shame, too, 'cause this place used to be fun...
QUOTE (Sapo+Sep 27 2008, 06:33 PM)
Likewise, any attempt at rational discussion here is pointless.
That is a shame, too, 'cause this place used to be fun...
What do you mean 'used to be'? Some of us are still having a ball.
That is a shame, too, 'cause this place used to be fun...
What do you mean 'used to be'? Some of us are still having a ball.
============================================================
B ) "Relatively prime" numbers are two numbers that share no common factors. In similar, but different vein, is there a name for the set of all numbers sharing the same number of divisors, none of which can be factors of or multiples of any other number in that set?
============================================================
In the seeming absence of any pre-existing term to apply to the sets of numbers referred to above, I have taken to calling any collection of integers sharing a common divisor number...
========================
Relatively Non-Composite Numbers
========================
... and they can be constructed into geometric sets and compared to the set of all prime numbers.
The set of all numbers with 12 divisors, for instance, constructed from any given number of primes, z, is equal to z * ((z^2 + z)/2), or, in other notation z*T_z, where T = Triangular Number.
The RATIO of constructible integers from z primes that have 12 divisors, relative to the set of primes with z elements is:
T_z
Extrapolate out to infinity, and well... I won't use words like "Triangular Infinity" lest I run afoul of the mathematical dogma that runs rampant upon this forum, but from a common sense perspective, it would sure look a lot like that.
But as I wrote, I won't use that term, and in fact, instead of finding the limit at infinity, let's let z = some big number, some number as big, in fact, as the mind can imagine, a number somewhere in the stratospheric outer reaches of the Numberverse where Skewe's Number (10^10^10^34) might well be called a neighborhood friend.
Out there in that realm of "virtual infinity," where Gauss' guess of the number of primes up to pi(n) and the number of primes at pi(n) finally converge (if the Riemann Hypthesis is true, that is), you would would have a set of relatively indivisible numbers the size of which would dwarf the primes no less so than the entirety of all the numbers in Pascal's Triangle dwarfs it's first element, the number 1 at the top of the pyramid.
Others might call that "trivial" or "pointless," but I don't, especially not when combining together these Relatively Non-Composite Sets I have been building gives me number progressions such as the following:
1, 3, 15, 84, 495, 3003 ...
... a series that can be found here:
Gauss' Hypergeometric Function:
2F1([1/3, 2/3], [1/2], 27 z/4)
http://www.lacim.uqam.ca/~plouffe/articles...eneratrices.pdf
When one finds oneself "accidentally" creating progressions related to such as the above, it gives one pause. And it makes one wonder if maybe, just maybe, one isn't on to something just a bit interesting.
Best,
Raphie
Hypergeometric Series
http://en.wikipedia.org/wiki/Hypergeometric_series
The Really Big Numbers Page
http://varatek.com/scott/bnum.html
B ) "Relatively prime" numbers are two numbers that share no common factors. In similar, but different vein, is there a name for the set of all numbers sharing the same number of divisors, none of which can be factors of or multiples of any other number in that set?
============================================================
In the seeming absence of any pre-existing term to apply to the sets of numbers referred to above, I have taken to calling any collection of integers sharing a common divisor number...
========================
Relatively Non-Composite Numbers
========================
... and they can be constructed into geometric sets and compared to the set of all prime numbers.
The set of all numbers with 12 divisors, for instance, constructed from any given number of primes, z, is equal to z * ((z^2 + z)/2), or, in other notation z*T_z, where T = Triangular Number.
The RATIO of constructible integers from z primes that have 12 divisors, relative to the set of primes with z elements is:
T_z
Extrapolate out to infinity, and well... I won't use words like "Triangular Infinity" lest I run afoul of the mathematical dogma that runs rampant upon this forum, but from a common sense perspective, it would sure look a lot like that.
But as I wrote, I won't use that term, and in fact, instead of finding the limit at infinity, let's let z = some big number, some number as big, in fact, as the mind can imagine, a number somewhere in the stratospheric outer reaches of the Numberverse where Skewe's Number (10^10^10^34) might well be called a neighborhood friend.
Out there in that realm of "virtual infinity," where Gauss' guess of the number of primes up to pi(n) and the number of primes at pi(n) finally converge (if the Riemann Hypthesis is true, that is), you would would have a set of relatively indivisible numbers the size of which would dwarf the primes no less so than the entirety of all the numbers in Pascal's Triangle dwarfs it's first element, the number 1 at the top of the pyramid.
Others might call that "trivial" or "pointless," but I don't, especially not when combining together these Relatively Non-Composite Sets I have been building gives me number progressions such as the following:
1, 3, 15, 84, 495, 3003 ...
... a series that can be found here:
Gauss' Hypergeometric Function:
2F1([1/3, 2/3], [1/2], 27 z/4)
http://www.lacim.uqam.ca/~plouffe/articles...eneratrices.pdf
When one finds oneself "accidentally" creating progressions related to such as the above, it gives one pause. And it makes one wonder if maybe, just maybe, one isn't on to something just a bit interesting.
Best,
Raphie
Hypergeometric Series
http://en.wikipedia.org/wiki/Hypergeometric_series
The Really Big Numbers Page
http://varatek.com/scott/bnum.html
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