Problem:
Prove that if 2n+1 and 3n+1 are both squares, then 5n+3 is not a prime.
Solution:
Proof by contradiction. Suppose that there is n such that 5n+3 is prime.
Note that 4(2n+1) - (3n+1) = 5n+3 .
Let 2n+1 = p^2 and 3n+1 = q^2 , where p, q > 0 . We have (2p-q)(2p+q) = 5n+3 . Since RHS is a prime, we must have 2p - q = 1 and 2p + q = 5n+3 . Solving for q we get 2q = 5n + 2 . Substituting, we get 2q = q^2 + 2n + 1 , or -2n = (q-1)^2 . Since RHS is \geq 0 , we can only have equality when n= 0 and q = 1 . In this case, we have 5n + 3 = 8 not a prime. QED.
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