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)