2008 IMO Problems/Problem 3
(still editing...)
The main idea is to take a gaussian prime and multiply it by a "smaller"
to get
. The rest is just making up the little details.
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. Let's see what happens when both
and
.
Since and
are apparently co-prime, there must exist integers
and
such that
\[ad+bc=1 \reqno{(1)}\] (Error compiling LaTeX. Unknown error_msg)
In fact, if and
are such numbers, then
and
work as well, so we can assume that
.
Define and let's see what happens. Notice that
.
If , then from (1), we get
and hence
. That means that