# Lucas's theorem

In number theory, Lucas's theorem expresses the remainder of division of the binomial coefficient ${\tbinom {m}{n}}$ by a prime number p in terms of the base p expansions of the integers m and n.
Lucas's theorem first appeared in 1878 in papers by Édouard Lucas.

## Statement

For non-negative integers m and n and a prime p, the following congruence relation holds:
    ${\binom {m}{n}}\equiv \prod _{i=0}^{k}{\binom {m_{i}}{n_{i}}}{\pmod {p}},$
where
    $m=m_{k}p^{k}+m_{k-1}p^{k-1}+\cdots +m_{1}p+m_{0},$
and
    $n=n_{k}p^{k}+n_{k-1}p^{k-1}+\cdots +n_{1}p+n_{0}$
are the base p expansions of m and n respectively. This uses the convention that ${\tbinom {m}{n}}=0$ if m < n.

## Proofs

There are several ways to prove Lucas's theorem.
Let M be a set with m elements, and divide it into m$_i$ cycles of length p$^i$ for the various values of i. Then each of these cycles can be rotated separately, so that a group G which is the Cartesian product of cyclic groups C$_{pi}$ acts on M. It thus also acts on subsets N of size n. Since the number of elements in G is a power of p, the same is true of any of its orbits. Thus in order to compute ${\tbinom {m}{n}}$ modulo p, we only need to consider fixed points of this group action. The fixed points are those subsets N that are a union of some of the cycles. More precisely one can show by induction on k-i, that N must have exactly n$_i$ cycles of size p$^i$. Thus the number of choices for N is exactly $\prod _{i=0}^{k}{\binom {m_{i}}{n_{i}}}{\pmod {p}}$.
This proof is due to Nathan Fine.
If p is a prime and n is an integer with 1 ≤ n ≤ p − 1, then the numerator of the binomial coefficient
    ${\binom {p}{n}}={\frac {p\cdot (p-1)\cdots (p-n+1)}{n\cdot (n-1)\cdots 1}}$
is divisible by p but the denominator is not. Hence p divides ${\tbinom {p}{n}}$. In terms of ordinary generating functions, this means that
    $(1+X)^{p}\equiv 1+X^{p}{\pmod {p}}.$
Continuing by induction, we have for every nonnegative integer i that
    $(1+X)^{p^{i}}\equiv 1+X^{p^{i}}{\pmod {p}}.$
Now let m be a nonnegative integer, and let p be a prime. Write m in base p, so that $m=\sum _{i=0}^{k}m_{i}p^{i}$ for some nonnegative integer k and integers m$_i$ with 0 ≤ m$_i$ ≤ p-1. Then
    ${\begin{aligned}\sum _{n=0}^{m}{\binom {m}{n}}X^{n}&=(1+X)^{m}=\prod _{i=0}^{k}\left((1+X)^{p^{i}}\right)^{m_{i}}\\&\equiv \prod _{i=0}^{k}\left(1+X^{p^{i}}\right)^{m_{i}}=\prod _{i=0}^{k}\left(\sum _{j_{i}=0}^{m_{i}}{\binom {m_{i}}{j_{i}}}X^{j_{i}p^{i}}\right)\\&=\prod _{i=0}^{k}\left(\sum _{j_{i}=0}^{p-1}{\binom {m_{i}}{j_{i}}}X^{j_{i}p^{i}}\right)\\&=\sum _{n=0}^{m}\left(\prod _{i=0}^{k}{\binom {m_{i}}{n_{i}}}\right)X^{n}{\pmod {p}},\end{aligned}}$
as the representation of n in base p is unique and in the final product, n$_i$ is the ith digit in the base p representation of n. This proves Lucas's theorem.

## Consequences

* A binomial coefficient ${\tbinom {m}{n}}$ is divisible by a prime p if and only if at least one of the base p digits of n is greater than the corresponding digit of m.
* In particular, ${\tbinom {m}{n}}$ is odd if and only if the binary digits (bits) in the binary expansion of n are a subset of the bits of m.

## Non-prime modulo

Lucas's theorem can be generalized to give an expression for the remainder when ${\tbinom {m}{n}}$ is divided by a prime power p$^k$. However, the formulas become more complicated.
If the modulo is the square of a prime p, the following congruence relation holds for all 0 ≤ s ≤ r ≤ p − 1, a ≥ 0, and b ≥ 0.
    ${\binom {pa+r}{pb+s}}\equiv {\binom {a}{b}}{\binom {r}{s}}(1+pa(H_{r}-H_{r-s})+pb(H_{r-s}-H_{s})){\pmod {p^{2}}},$
where $H_{n}=1+{\tfrac {1}{2}}+{\tfrac {1}{3}}+\cdots +{\tfrac {1}{n}}$ is the nth harmonic number.
Generalizations of Lucas's theorem for higher prime powers p$^k$ are also given by Davis and Webb (1990) and Granville (1997).

## Variations and generalizations

* Kummer's theorem asserts that the largest integer k such that p$^k$ divides the binomial coefficient ${\tbinom {m}{n}}$ (or in other words, the valuation of the binomial coefficient with respect to the prime p) is equal to the number of carries that occur when n and m − n are added in the base p.
* The q-Lucas theorem is a generalization for the q-binomial coefficients, first proved by J. Désarménien.