Factorial - Number Theory

Number Theory

Factorials have many applications in number theory. In particular, n! is necessarily divisible by all prime numbers up to and including n. As a consequence, n > 5 is a composite number if and only if

A stronger result is Wilson's theorem, which states that

if and only if p is prime.

Adrien-Marie Legendre found that the multiplicity of the prime p occurring in the prime factorization of n! can be expressed exactly as

This fact is based on counting the number of factors p of the integers from 1 to n. The number of multiples of p in the numbers 1 to n are given by ; however, this formula counts those numbers with two factors of p only once. Hence another factors of p must be counted too. Similarly for three, four, five factors, to infinity. The sum is finite since p i can only be less than or equal to n for finitely many values of i, and the floor function results in 0 when applied for p i > n.

The only factorial that is also a prime number is 2, but there are many primes of the form n! ± 1, called factorial primes.

All factorials greater than 1! are even, as they are all multiples of 2. Also, all factorials from 5! upwards are multiples of 10 (and hence have a trailing zero as their final digit), because they are multiples of 5 and 2.

Also note that the reciprocals of factorials produce a convergent series: (see e)