Legendre's formula

In mathematics, Legendre's formula gives an expression for the exponent of the largest power of a prime p that divides the factorial n!. It is named for Adrien-Marie Legendre. It is also sometimes known as de Polignac's formula, after Alphonse de Polignac.

For any prime number p and any integer n, let ${\displaystyle \nu _{p}(n)}$ be the exponent of the largest power of p that divides n (that is, the p-adic valuation of n). Then

where ${\displaystyle \lfloor x\rfloor }$ is the floor function. While the formula on the right side is an infinite sum, for any particular values of n and p it has only finitely many nonzero terms: for every i large enough that ${\displaystyle p^{i}>n}$, one has ${\displaystyle \textstyle \left\lfloor {\frac {n}{p^{i}}}\right\rfloor =0}$.

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