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 ${\displaystyle \nu _{p}(n)}$ be the exponent of the largest power of p that divides n (that is, the  p-adic valuation of n). Then

${\displaystyle \nu _{p}(n!)=\sum _{i=1}^{\infty }\left\lfloor {\frac {n}{p^{i}}}\right\rfloor ,}$

where ${\displaystyle \lfloor x\rfloor }$ 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 ${\displaystyle p^{i}>n}$, one has ${\displaystyle \textstyle \left\lfloor {\frac {n}{p^{i}}}\right\rfloor =0}$. This reduces the infinite sum above to

${\displaystyle \nu _{p}(n!)=\sum _{i=1}^{L}\left\lfloor {\frac {n}{p^{i}}}\right\rfloor \,,}$

where ${\displaystyle L=\lfloor \log _{p}n\rfloor }$.

Example

For n = 6, one has ${\displaystyle 6!=720=2^{4}\cdot 3^{2}\cdot 5^{1}}$. The exponents ${\displaystyle \nu _{2}(6!)=4,\nu _{3}(6!)=2}$ and ${\displaystyle \nu _{5}(6!)=1}$ can be computed by Legendre's formula as follows:

${\displaystyle {\begin{aligned}\nu _{2}(6!)&=\sum _{i=1}^{\infty }\left\lfloor {\frac {6}{2^{i}}}\right\rfloor =\left\lfloor {\frac {6}{2}}\right\rfloor +\left\lfloor {\frac {6}{4}}\right\rfloor =3+1=4,\\[3pt]\nu _{3}(6!)&=\sum _{i=1}^{\infty }\left\lfloor {\frac {6}{3^{i}}}\right\rfloor =\left\lfloor {\frac {6}{3}}\right\rfloor =2,\\[3pt]\nu _{5}(6!)&=\sum _{i=1}^{\infty }\left\lfloor {\frac {6}{5^{i}}}\right\rfloor =\left\lfloor {\frac {6}{5}}\right\rfloor =1.\end{aligned}}}$

Proof

Since ${\displaystyle n!}$ is the product of the integers 1 through n, we obtain at least one factor of p in ${\displaystyle n!}$ for each multiple of p in ${\displaystyle \{1,2,\dots ,n\}}$, of which there are ${\displaystyle \textstyle \left\lfloor {\frac {n}{p}}\right\rfloor }$. Each multiple of ${\displaystyle p^{2}}$ contributes an additional factor of p, each multiple of ${\displaystyle p^{3}}$ contributes yet another factor of p, etc. Adding up the number of these factors gives the infinite sum for ${\displaystyle \nu _{p}(n!)}$.

Alternate form

One may also reformulate Legendre's formula in terms of the base-p expansion of n. Let ${\displaystyle s_{p}(n)}$ denote the sum of the digits in the base-p expansion of n; then

${\displaystyle \nu _{p}(n!)={\frac {n-s_{p}(n)}{p-1}}.}$

For example, writing n = 6 in binary as 6₁₀ = 110₂, we have that ${\displaystyle s_{2}(6)=1+1+0=2}$ and so

${\displaystyle \nu _{2}(6!)={\frac {6-2}{2-1}}=4.}$

Similarly, writing 6 in ternary as 6₁₀ = 20₃, we have that ${\displaystyle s_{3}(6)=2+0=2}$ and so

${\displaystyle \nu _{3}(6!)={\frac {6-2}{3-1}}=2.}$

Proof

Write ${\displaystyle n=n_{\ell }p^{\ell }+\cdots +n_{1}p+n_{0}}$ in base p. Then ${\displaystyle \textstyle \left\lfloor {\frac {n}{p^{i}}}\right\rfloor =n_{\ell }p^{\ell -i}+\cdots +n_{i+1}p+n_{i}}$, and therefore

${\displaystyle {\begin{aligned}\nu _{p}(n!)&=\sum _{i=1}^{\ell }\left\lfloor {\frac {n}{p^{i}}}\right\rfloor \\&=\sum _{i=1}^{\ell }\left(n_{\ell }p^{\ell -i}+\cdots +n_{i+1}p+n_{i}\right)\\&=\sum _{i=1}^{\ell }\sum _{j=i}^{\ell }n_{j}p^{j-i}\\&=\sum _{j=1}^{\ell }\sum _{i=1}^{j}n_{j}p^{j-i}\\&=\sum _{j=1}^{\ell }n_{j}\cdot {\frac {p^{j}-1}{p-1}}\\&=\sum _{j=0}^{\ell }n_{j}\cdot {\frac {p^{j}-1}{p-1}}\\&={\frac {1}{p-1}}\sum _{j=0}^{\ell }\left(n_{j}p^{j}-n_{j}\right)\\&={\frac {1}{p-1}}\left(n-s_{p}(n)\right).\end{aligned}}}$

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 ${\displaystyle {\binom {2n}{n}}}$ 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 ${\displaystyle p^{-1/(p-1)}}$.

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.

External links

• Weisstein, Eric W. "Factorial". MathWorld.

This page was last edited on 10 May 2024, at 12:34