2008 IMO Problems/Problem 3
(still editing...)
The main idea is to take a gaussian prime and multiply it by a "twice 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
. Apparently,
, so assume without loss of generality that
.
Since and
are (obviously) co-prime, there must exist integers
and
such that
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 $a\2$ (Error compiling LaTeX. Unknown error_msg) and hence
. That means that
and
. Therefore,
and so
, from where
. Finally,
and the case
is cleared.
We can safely assume now that
Automatically,
, since
since
implies
.