Mersenne Conjecture

[Gammatester]

Although English is not my native langage, I think your Proposition 1:

If M_p is not prime, when this is true: phi(M_p) = 0 mod p^2

should read

If M_p is not prime, then this is true: phi(M_p) = 0 mod p^2.

But M_4 = 15 is a counterexample because phi(15) mod 16 = 8. Or am I missing something here, e.g. that p should be prime?

[R.D. Silverman]

Originally Posted by Gammatester

Although English is not my native langage, I think your Proposition 1:

If M_p is not prime, when this is true: phi(M_p) = 0 mod p^2

should read

If M_p is not prime, then this is true: phi(M_p) = 0 mod p^2.

But M_4 = 15 is a counterexample because phi(15) mod 16 = 8. Or am I missing something here, e.g. that p should be prime?

If p is prime, the conjecture is true and TRIVIAL.

Given N = 2^p-1, and N is composite then N is the product of at least two
primes, each of which is 1 mod p. phi(N) will be divisible by p^k, where
k is the number of distinct prime factors of N.

This does not merit calling it a 'conjecture'. It is an elementary homework
problem that one might assign to a beginning number theory class.

[sascha77]

Originally Posted by R.D. Silverman

If p is prime, the conjecture is true and TRIVIAL.

Given N = 2^p-1, and N is composite then N is the product of at least two
primes, each of which is 1 mod p. phi(N) will be divisible by p^k, where
k is the number of distinct prime factors of N.

This does not merit calling it a 'conjecture'. It is an elementary homework
problem that one might assign to a beginning number theory class.

Hello Silverman, Yes i mean that p must be Prime. M_p is the abbreviation of Mersenne-number with p prime. Thank you, that you showed me, that Proposition 1 is trivial. But it was not the conjecture , the conjecture is in top of the pdf. (Conjecture 1) And I showed that proposition1 is true, because i need this for example in proposition5. And I wasnt shure if everybody see the Proposition1 is trivial. So I simply showed that this is true. I have question to you: Let M_p be a Mersenne-PRIME Number (M_p is Prime). When it has the form : M_p = 2^p-1 = 2*k*p+1 , with k is natural number. Can you show that k==0 (mod p) is always false. Perhaps it is also trivial, I do not know. Sascha

[R.D. Silverman]

Originally Posted by sascha77

Hello Silverman, Yes i mean that p must be Prime. M_p is the abbreviation of Mersenne-number with p prime. Thank you, that you showed me, that Proposition 1 is trivial. But it was not the conjecture , the conjecture is in top of the pdf. (Conjecture 1) And I showed that proposition1 is true, because i need this for example in proposition5. And I wasnt shure if everybody see the Proposition1 is trivial. So I simply showed that this is true. I have question to you: Let M_p be a Mersenne-PRIME Number (M_p is Prime). When it has the form : M_p = 2^p-1 = 2*k*p+1 , with k is natural number. Can you show that k==0 (mod p) is always false. Perhaps it is also trivial, I do not know. Sascha

Repeat after me: Google is my friend.

Look up "Wieferich"

[cheesehead]

Also, Mathworld (http://mathworld.wolfram.com/) is your mathematical friend, usually.

http://mathworld.wolfram.com/search/...erich&x=10&y=9