20101215, 15:16  #1 
Sep 2009
2^{2}×3^{2} Posts 
Properties of Mersenne numbers
In Tony Reix's Properties of Mersenne and Fermat numbers online paper
you see: Mq is a prime if and only if there exists only one pair (x, y) such that: Mq = (2x)^2+ 3(3y)^2. The proof is missing. Can anybody provide a proof? By numerical testing different Mq values I have found that if Mq is composite there is no pair (x,y) that satisfies the condition. Is it possible that if Mq is composite there can be 2 or more pairs? Thanx in advance... 
20101215, 15:35  #2  
"Bob Silverman"
Nov 2003
North of Boston
2^{2}·1,877 Posts 
Quote:
I will sketch a proof. This result has very little to do with Mersenne primes. Let Q = (2x)^2 + 3(3y)^2. Q is prime iff this representation is unique. Now, follow the (standard!) proof that an integer that is 1 mod 4 is prime iff it is the sum of two squares in a unique way. i.e. Factor Q over Q(sqrt(3)) and observe that you are doing so in a UFD. [QUOTE] 

20101215, 21:20  #3  
May 2004
New York City
5·7·11^{2} Posts 
Quote:


20101216, 16:06  #4  
Sep 2009
2^{2}·3^{2} Posts 
Quote:
I'm stuck. If there is a unique pair (x,y) then Q is prime,however , if Q is composite, then can we assume that there are no (x,y) pairs or should we consider there are 2,3 or more pairs? Thanks Last fiddled with by wblipp on 20101216 at 19:08 Reason: fix quotes 

20101216, 18:43  #5  
"Bob Silverman"
Nov 2003
North of Boston
2^{2}×1,877 Posts 
[QUOTE=kurtulmehtap;242188]
Quote:


20101217, 19:41  #6 
May 2004
New York City
5×7×11^{2} Posts 

20101223, 21:48  #7 
May 2004
New York City
5·7·11^{2} Posts 
So is the OPer satisfied?

20110105, 14:15  #8 
Sep 2009
2^{2}×3^{2} Posts 
Not Really, I am still not sure if a composite Mersenne number can have more than 1 pair for x^2 + 27y^2.
There is a thesis on this subject: Mersenne primes of the form x^2+dy^2 by Bas Jansen at www.math.leidenuniv.nl/en/theses/31/ It has an entire section for the needed case d=27, but I still can't find the answer.. Please help. I know that I am embarassing myself but I need the answer. 
20110105, 15:20  #9  
"Bob Silverman"
Nov 2003
North of Boston
2^{2}·1,877 Posts 
Quote:


20110105, 15:22  #10 
"Bob Silverman"
Nov 2003
North of Boston
2^{2}·1,877 Posts 

20110105, 15:23  #11 
"Forget I exist"
Jul 2009
Dartmouth NS
2×3×23×61 Posts 

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