Factorial - Imported Knowlpedia Content

For a natural number $n$ , the factorial of $n$ is

n!=\prod_{j=1}^{n}j=1\cdot2\cdots n,

with the empty-product convention $0!=1$ . It satisfies the recursion $(n+1)!=(n+1)n!$ .

Factorials encode the prime multiplicities accumulated in an initial interval. For a prime $p$ , Legendre's formula gives

v_p(n!)=\sum_{r\ge1}\left\lfloor\frac{n}{p^r}\right\rfloor,

where only finitely many summands are nonzero and $v_p$ is the $p$ -adic valuation.