New Mersenne Primes Characterizacion

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• Hello:   I send you a theorem that I have found yesterday   New Mersenne Primes Characterization.   Theorem:   Let p a  odd prime number.   If exist k
Message 1 of 1 , May 10, 2009
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Hello:

I send you a theorem that I have found yesterday

New Mersenne Primes Characterization.

Theorem:

Let p a  odd prime number.

If exist k  positive integer that Sqrt[p^2 + 2^(k+2)] is integer then are verified:

1) Sqrt[p^2 + 2^(k+2)]=p+2
2) k is prime
3) p is Mersenne prime p=Mk

Proof:

1)
Sqrt[p^2 + 2^(k+2)] is integer then
Sqrt[p^2 + 2^(k+2)]=p+c  with c positive integer then
then p^2+2^(k+2)=p^2+2cp+c^2  --> 2^(k+2)=c(2p+c)  -->
c divides 2^(k+2)  --> c=2^t

Then: p^2+2^(k+2)=p^2+2^(t+1)p+2^(2t)   since k+2>t+1 then:

p odd -->
2^(k+2-t-1)=p+2^(2t-t-1)  --> p+2^(2t-t-1) is even --> 2t-t-1=0 -->
t=1--> c=2^1=2

2)

p^2+2^(k+2)=(p+2)^2 --> p^2+2^(k+2)=p^2+4p+4--> 2^k=p+1-->p=2^k-1
p is prime then k is prime.

3)

Trivial by 2) p=Mk

Sincerely

Sebastian Martin Ruiz

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