Quote:
Originally Posted by kurtulmehtap
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?

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.
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