4 ms·
>For every n, there is some base b such that n! = b^n. For example, 30! ≈ 12^30
by warrenm 3y ago
>For every n, there is some base b such that n! = b^n. For example, 30! ≈ 12^30
- colanderman 3y agoI think missing here is that "=" should be a "≈", and that "b" should be an integer. Otherwise this is trivially false (consider 2!=b^2 if b must be integer) or true (if b need not be integer).
- eesmith 3y agoIt is the trivial case, b = exp(log(n!)/n) 30! = 12.04449703813164.... ** 30 It's leading on to the next part: "What’s interesting is that b is very nearly a linear function of n. In hindsight it’s clear that this should be the case—it follows easily from Stirling’s approximation—but I didn’t anticipate this before I plotted it."