How big are factorials?

Stirling's approximation is also used a lot in statistical mechanics, because you often have to calculate logs of state space sizes, which means lots of combinatorics and thus lots of factorials. Plus it's continuous so you can do calculus.

Re: How big are factorials?

#5

Post by ninju » Wed, Sep 16, 2026, 3:05 PM UTC

The author's casual mention of 52! at the opening of the article triggered an OLD webpage that I saw many years ago

https://czep.net/weblog/52cards.html

Anyone know how to determine the age of this page (it's got be at least 20yrs old)

Re: How big are factorials?

#6

Post by Sharlin » Wed, Sep 16, 2026, 3:29 PM UTC

A quick and dirty approximation of the number of digits in n! is n lg n, which approximates n! from above, via the inequality

  1 * 2 * … * n ≤ n * … * n.

(This approximation should be familiar to many from an algorithmics class.)

For a tighter bound, use n lg n - n/2, or a better approximation of ln 10 in place of 1/2 if you wish. This comes from Stirling's approximation which notes that

  ln n! = n ln n - n + O(ln n).

Re: How big are factorials?

#7

Post by TheRealPomax » Wed, Sep 16, 2026, 3:36 PM UTC

ninju wrote: post #5

The author's casual mention of 52! at the opening of the article triggered an OLD webpage that I saw many years ago https://czep.net/weblog/52cards.html Anyone know how to determine the age of this page (it's got be at least 20yrs old)

The main.css file it imports dates itself to March 9 of 2005, and is housed in an "ancient history" section of the website that covers everything before October 26, 2010, so: "sometime between those two years" =P

Re: How big are factorials?

#8

Post by pkaye » Wed, Sep 16, 2026, 3:42 PM UTC

pagade wrote: post #3

Reminds me of: Professor asked us to find the biggest factorial using C programming language. And then using LISP. You can imagine our surprise.

There is a algorithm call Prime Swing Factorial that can compute large factorials exactly in arbitrary precision math using prime factorization. Like 10000000! in under second depending of how optimized the math library it. Probably like 100x faster than the normal method.

Re: How big are factorials?

#9

Post by stronglikedan » Wed, Sep 16, 2026, 3:43 PM UTC

ninju wrote: post #5

The author's casual mention of 52! at the opening of the article triggered an OLD webpage that I saw many years ago https://czep.net/weblog/52cards.html Anyone know how to determine the age of this page (it's got be at least 20yrs old)

52 cards is the first thing I think of when I think factorials. It's such a great and relatable way to convey the subject to people, plus it usually ends up blowing their minds like it did mine when I first learned of it. Not from this page, but from a YT vid many moons ago.

Re: How big are factorials?

#10

Post by Dwedit » Wed, Sep 16, 2026, 3:47 PM UTC

ninju wrote: post #5

The author's casual mention of 52! at the opening of the article triggered an OLD webpage that I saw many years ago https://czep.net/weblog/52cards.html Anyone know how to determine the age of this page (it's got be at least 20yrs old)

It was made during the brief XHTML craze. (And it's also invalid XHTML)