brocard | unsolved problem in mathematical number theory | Learning library

Your algorithm is O(n^3).

You have two nested loops, and inside you use factorial(), having O(n) complexity itself.

Your algorithm tests all (n,m) combinations, even those where factorial(n) and m^2 are far apart, e.g. n=1 and m=10000.

You always recompute the factorial(n) deep inside the loop, although it's independent of the inner loop variable m. So, it could be moved outside of the inner loop.

And, instead of always computing factorial(n) from scratch, you could do that incrementally. Whenever you increment n by 1, you can multiply the previous factorial by n.

A different, better approach would be not to use nested loops, but to always keep n and m in a number range so that factorial(n) is close to m^2, to avoid checking number pairs that are vastly off. We can do this by deciding which variable to increment next. If the factorial is smaller, then the next brocard pair needs a bigger n. If the square is smaller, we need a bigger m.

In pseudo code, that would be