I don't know, other than he was very smart. But a potential way that the article hints at is that the factorial is related to the gamma function which I believe

I don't know, other than he was very smart. But a potential way that the article hints at is that the factorial is related to the gamma function which I believe is the solution to a differential equation... and so there may be some insight to be had in looking at the relevant differential equation and trying to approximate its solution. Edit after googling: the gamma function not the solution to a differential equation, in fact there is a proof that it cant be [*] but it's formulated as an integral, whose approximation could lead to odd looking equations like the one in the article https://en.m.wikipedia.org/wiki/H%C3%B6lder%27s_theorem https://en.m.wikipedia.org/wiki/H%C3%B6lder%27s_theorem