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Find prime p, q, s.t. p^2 - 2q^2 = 1

Problem:
Find prime p, q which satisfies p^2 - 2q^2 = 1.

Solution:
We see that p = 3, q = 2 is a solution. We want to show that this is the only solution.

Suppose there exists a solution with q > 2 . Rearranging the equation we have (p-1)(p+1) = 2q^2 . Since RHS is divisible by 2, we know that p must be an odd prime. Hence, both p-1 and p+1 are divisible by 2. That means LHS is at least divisible by 4. Hence 2q^2 must be divisible by 4. This is only possible when 2 | q . Contradiction.

QED

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