2008 IMO Problems/Problem 3

(still editing...)\ For each sufficiently large prime $p$ of the form $4k+1$ , we shall find a corresponding $n$ satisfying the required condition with the prime number in question being $p$ . Since there exist infinitely many such primes and, for each of them, $n \ge \sqrt(p-1)$ , we will have found infinitely many distinct $n$ satisfying the problem.