2022 USAJMO Problems/Problem 5
Problem
Find all pairs of primes for which and are both perfect squares.
Solution
Let , , where are positive integers. . So,
For , . Then and . and . Thus, and we find . Hence .
For , ( integer), by , . Let's examine in , . But we know that . This is a contradiction and no solution for .
For , ( integer), by , . Let , where and are integers. Since , we see . Thus, by , . and are same parity and is even integer. So, and are both even integers. Therefore,
or Therefore, or . For each case, . But , this gives a contradiction. No solution for .
We conclude that the only solution is .
(Lokman GÖKÇE)