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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