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 .