# 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* (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 .

### 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 6_{10} = 110_{2}, we have that and so

Similarly, writing 6 in ternary as 6_{10} = 20_{3}, 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 .

## References

- 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.