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?