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
See Also
2022 USAJMO (Problems • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |
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