Approaches to Brocard's problem

Question

The brocard problem is an unsolved problem that asks how many integers m and n exist such that

$n! + 1 = m²$

More specifically it conjectures there are only three such numbers. What are some interesting ways to approach this conjecture? I've being playing around with it for over an hour and I tried the following approach.

For any n, what is the smallest integer a, such that $n! + a = m²$? I created a function $v(n!)=a$ and now restated the conjecture in the following terms. $ v(n!) > 0 $ for all n except 4,5,7

I checked to see if v(n!) had any interesting pattern and as expected I found none. The next natural thing I wanted to do was to graph the function. To do this I need to somehow make n!, naturally we can do this with the gamma function. $v(\gamma(n))$. I am no expert on the gamma function, but I'd be really interested to see what patterns could emerge from such extensions. Do you know any papers/books discussing interesting approaches to resolving this conjecture?

Answer 1

Hello Ryan I have been interested in this problem due to its simplicity. Your first question. I have preferred to solve it elementary with bounds. For $m \lt 5$ $m!$ ends in a $0$. Therefore $m! +1$ ends in a 1. Therefore $m!+1= (10x+1)^2)$ or $ m!+1=(10x+9)^2)$ Expand and simplify. $100x^2+20x=m!$ @fleablood told us about the factorisation of $m!/20=x (5x+1)$ and that $x$ and $5x+1$ are coprime.