View Single Post
2010-12-16, 18:43   #5
R.D. Silverman

Nov 2003

22·5·373 Posts

[QUOTE=kurtulmehtap;242188]
Quote:
 Originally Posted by R.D. Silverman I have not verified that the result is true. I will assume that it is. 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. Thanks a lot. 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