Any number can start a factorial

= 2019…000 The factorial of 3177 is a number with 9749 digits, the first of which are 2019, and the last 793 of which are zeros.

The solution m = 3177 was only the first.

The next solution is 6878, and there are infinitely more.

Not only does every number appear at the beginning of a factorial, it appears at the beginning of infinitely many factorials.

We can say even more.

Persi Diaconis proved that factorials obey Benford’s law, and so we can say how often a number n appears at the beginning of a factorial.

If a sequence of numbers, like factorials, obeys Benford’s law, then the leading digit d in base b appears with probability logb(1 + 1/d).

If we’re interested in 4-digit numbers like 2019, we can think of these as base 10,000 digits.

This means that the proportion factorials that begin with 2019 equals log10000(1 + 1/2019) or about 1 in every 18,600 factorials.

By the way, powers of 2 also obey Benford’s law, and so you can find any number at the beginning of a power of 2.

For example, 22044 = 2019… Related: Benford’s law blog posts.

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