2022 USAJMO Problems/Problem 5
Problem
Find all pairs of primes for which and are both perfect squares.
Solution 1
We first consider the case where one of is even. If , and which doesn't satisfy the problem restraints. If , we can set and giving us . This forces so giving us the solution .
Now assume that are both odd primes. Set and so . Since , . Note that is an even integer and since and have the same parity, they both must be even. Therefore, for some positive even integer . On the other hand, and . Therefore, so , giving us a contradiction.
Therefore, the only solution to this problem is .
~BennettHuang