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 after Adrien-Marie Legendre. It is also sometimes known as de Polignac's formula, after Alphonse de Polignac.

Statement

For any prime number p and any positive integer n, let be the exponent of the largest power of p that divides n. Then
where is the floor function. While the sum 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, one has. This reduces the infinite sum above to
where.

Example

For n = 6, one has. The exponents and can be computed by Legendre's formula as follows:

Proof

Since is the product of the integers 1 through n, we obtain at least one factor of p in for each multiple of p in, of which there are. Each multiple of contributes an additional factor of p, each multiple of contributes yet another factor of p, etc. Adding up the number of these factors gives the infinite sum for.

Alternate form

One may also reformulate Legendre's formula in terms of the base-p expansion of n. Let denote the sum of the digits in the base-p expansion of n; then
For example, writing n = 6 in binary as 6₁₀ = 110₂, we have that and so
Similarly, writing 6 in ternary as 6₁₀ = 20₃, we have that and so

Proof

Write in base p. Then, and therefore

Applications

Legendre's formula can be used to prove Kummer's theorem. As one special case, it can be used to prove that if n is a positive integer then 4 divides if and only if n is not a power of 2.
It follows from Legendre's formula that the p-adic exponential function has radius of convergence.