Re: [PrimeNumbers] (im)perfect numbers
- Sunday, December 05, 2004 11:19 PM [GMT+1=CET],
Michael Gian <work.gian@...> escribió:
> As I have read, for prime numbers, p, 2^p-1 are called Prime-The sum of the first n odd cubes is
> Exponent Mersenne numbers.
> These are sub-group of 2^k-1, k positive integers, which are called
> (just) Mersenne numbers.
> When 2^p-1 is prime, it is called a Mersenne prime.
> (2^(p-1)*(2^p-1)is a Perfect Number when 2^p-1 is a Mersenne prime.
> Does anyone know of a accepted term for a number of the form (2^(p-1)
> *(2^p-1) when 2^p-1 is not prime?
> The reason I ask is that all numbers of this form, including Perfect
> Numbers, are the sum of consecutive odd cubes. I am searching for a
> more concise way to talk about them.
S(n) = Sum((2k-1)^3, k, 1, n) = 2n^4 - n^2 = n^2(2n^2 - 1)
If n = 2^m, then
S(2^m) = 2^(2m)(2^(2m+1) - 1) = 2^(p-1)(2^p - 1) = M(p)
Then Mersenne number M(p) = 2^(p-1)(2^p - 1), with odd p, but no neccesary
prime, is the sum of the first 2^((p-1)/2) odd cubes.
Ignacio Larrosa Cañestro
A Coruña (España)