2008 IMO Problems/Problem 3
(still editing...)
For each sufficiently large prime of the form
, we shall find a corresponding
satisfying the required condition with the prime number in question being
. Since there exist infinitely many such primes and, for each of them,
, we will have found infinitely many distinct
satisfying the problem.
Take a prime of the form
and consider its "sum-of-two squares" representation
, which we know to exist for all such primes. If
or
, then
or
is our guy, and
as long as
(and hence
) is large enough.