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
.
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 $\ds d=-\frac{b-1}{2}$ (Error compiling LaTeX. Unknown error_msg) and
. Therefore,
and so
, from where $\ds b > \sqrt{2n}+\frac{1}{2}$ (Error compiling LaTeX. Unknown error_msg). Finally,
and the case $\ds c=\pm\frac{a}{2}$ (Error compiling LaTeX. Unknown error_msg) is cleared.
We can safely assume now that
Automatically,
.