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.
For any prime number p and any positive integer n, let
be the exponent of the largest power of p that divides n (that is, the p-adic valuation of n).
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
This reduces the infinite sum above to where
ν
ν
ν
can be computed by Legendre's formula as follows: 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
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
ν
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 Write
ℓ
ℓ
ℓ
ℓ − i
, and therefore 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
It follows from Legendre's formula that the p-adic exponential function has radius of convergence