Solution of Brocard’s Problem

Abstract

Brocard's problem is the solution of the equation,n! + 1=m2, where m and n are natural numbers. So far, only three solutions have been found, namely (n, m) = (4, 5), (5, 11), and (7, 71). The purpose of this paper is to show that there are no other solutions. Firstly, it will be shown that if (n,m) is to be a solution to Brocard's problem, then n! = 4AB, where A is even, B is odd, and |A – B| = 1. If n is even (n = 2x) and > 4, it will be shown that necessarily A = (2x)‼ 4y and B = y (2x-1)‼, for some odd y > 1. Next, it will be shown that x < 2y, and this leads to an inequality in x [namely, (x(2x − 1)‼ +- 1) 2 − 1 − (2x)! < 0], for which there is no solution when x ≥ 3. If n is odd, a similar procedure applies.