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). Then
where 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 , one has .
For n = 6, one has . The exponents and 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 , 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 .
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 in base p. Then , 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 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 .
- Legendre, A. M. (1830), Théorie des Nombres, Paris: Firmin Didot Frères
- Moll, Victor H. (2012), Numbers and Functions, American Mathematical Society, ISBN 978-0821887950, MR 2963308, page 77
- Leonard Eugene Dickson, History of the Theory of Numbers, Volume 1, Carnegie Institution of Washington, 1919, page 263.