Difference between revisions of "2022 USAMO Problems/Problem 4"
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==Problem== | ==Problem== | ||
+ | |||
+ | Find all pairs of primes <math>(p, q)</math> for which <math>p-q</math> and <math>pq-q</math> are both perfect squares. | ||
==Solution== | ==Solution== |
Revision as of 14:25, 15 April 2022
Problem
Find all pairs of primes for which
and
are both perfect squares.
Solution
Since is a perfect square and
is prime, we should have
for some positive integer
. Let
. Therefore,
, and substituting that into the
and solving for
gives
Notice that we also have
and so
. We run through the cases
: Then
so
, which works.
: This means
, so
, a contradiction.
: This means that
. Since
can be split up into two factors
such that
and
, we get
and each factor is greater than
, contradicting the primality of
.
Thus, the only solution is .
See also
2022 USAMO (Problems • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.