Let p and q be prime numbers such that q = 2p + 1, and p = 3 (mod 4). Prove that 2^p- 1 is a prime number (hence, a Mersenne prime!) if and only if p = 3. Hint. The "if" direction is easy. For the "only if" direction, proceed by proof by contradiction: by combining p= 3 and primality of 2^p 1 derive that q must divide 2^p - 1.

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