The number of digits in n! (Part 2)

The following graph shows the number of digits in n! as a function of n.

factorialdigits

When I was in school, I stared at this graph for weeks, if not months, trying to figure out an equation that would fit these points. And I never could figure it out.

In retrospect, my biggest mistake was thinking that the formula had to be something like y = a x^m, where the exponent m was a little larger than 1. After all, the graph is clearly not a straight line, but it’s also not as curved as a parabola.

What I didn’t know then, but know now, is that there’s a really easy way to determine to determine if a data set exhibits power-law behavior. If y = a x^m, then

\ln y = \ln a + m \ln x.

If we make the substitutions Y = \ln Y, B = \ln a, and X = \ln x, then this equation becomes

Y = m X + B

In other words, if the data exhibits power-law behavior, then the log-transformed data would look very much like a straight line. Well, here’s the graph of (X,Y) after applying the transformation:

loglogfactorialdigits

Ignoring the first couple of pots, the dots show an ever-so-slight concave down pattern, but not enough that would have discouraged me from blindly trying a pattern like y = a x^m. However, because these points do not lie on a straight line and exhibit heteroscedastic behavior, my adolescent self was doomed to failure.

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  1. The number of digits of n!: Index | Mean Green Math

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