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Old 2010-05-03, 14:04   #12
Gammatester
 
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Although English is not my native langage, I think your Proposition 1:
Code:
If M_p is not prime, when this is true: phi(M_p) = 0 mod p^2
should read
Code:
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?
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Old 2010-05-03, 14:25   #13
R.D. Silverman
 
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"Bob Silverman"
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Quote:
Originally Posted by Gammatester View Post
Although English is not my native langage, I think your Proposition 1:
Code:
If M_p is not prime, when this is true: phi(M_p) = 0 mod p^2
should read
Code:
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.
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Old 2010-05-07, 06:08   #14
sascha77
 
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Quote:
Originally Posted by R.D. Silverman View Post
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
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Old 2010-05-07, 11:09   #15
R.D. Silverman
 
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Quote:
Originally Posted by sascha77 View Post
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"
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Old 2010-05-08, 00:33   #16
cheesehead
 
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"Richard B. Woods"
Aug 2002
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Also, Mathworld (http://mathworld.wolfram.com/) is your mathematical friend, usually.

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