The previous post mentioned that 24! ≈ 10^{24} and 25! ≈ 10^{25}.

For every *n*, there is some base *b* such that *n*! = *b*^{n}. For example, 30! ≈ 12^{30}.

It’s easy to find *b*:

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.

Now fix *n* and find *b* such that *n*! = *b*^{n}. Since the relationship between *n* and *b*(*n*) is nearly linear, this suggests

which is true. It follows from the multiplication identity for the gamma function:

Let *z* = *n* + 1/2 so that the left side is (2*n*)!. On the right side, Γ(*z* + 1/2) = *n*! and Γ(*z*) is not too different from *n*!. The rest of the right side is 2^{2n}/√π.

So our observation that *b*(*n*) is nearly linear gave us a hint of Gauss’s multiplication formula.

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