Lucas' theorem

For the theorem in complex analysis, see Gauss–Lucas 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.

Formulation

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

$m=m_kp^k+m_{k-1}p^{k-1}+\cdots +m_1p+m_0,$

and

$n=n_kp^k+n_{k-1}p^{k-1}+\cdots +n_1p+n_0$

are the base p expansions of m and n respectively. This uses the convention that $\tbinom{m}{n}$ = 0 if m < n.

Consequence

• 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.

Proofs

There are several ways to prove Lucas's theorem. We first give a combinatorial argument and then a proof based on generating functions.

Combinatorial argument

Let M be a set with m elements, and divide it into m_i cycles of length pⁱ 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 pⁱ. Thus the number of choices for N is exactly $\prod_{i=0}^k\binom{m_i}{n_i}\pmod{p}$.

Proof based on generating functions

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

$(1+X)^p\equiv1+X^p\pmod{p}.$

Continuing by induction, we have for every nonnegative integer i that

$(1+X)^{p^i}\equiv1+X^{p^i}\pmod{p}.$

Now let m be a nonnegative integer, and let p be a prime. Write m in base p, so that $m=\sum_{i=0}^{k}m_ip^i$ for some nonnegative integer k and integers m_i with 0 ≤ m_i ≤ p-1. Then

$\begin{align} \sum_{n=0}^{m}\binom{m}{n}X^n & =(1+X)^m=\prod_{i=0}^{k}\left((1+X)^{p^i}\right)^{m_i}\\ & \equiv \prod_{i=0}^{k}\left(1+X^{p^i}\right)^{m_i} =\prod_{i=0}^{k}\left(\sum_{n_i=0}^{m_i}\binom{m_i}{n_i}X^{n_ip^i}\right)\\ & =\prod_{i=0}^{k}\left(\sum_{n_i=0}^{p-1}\binom{m_i}{n_i}X^{n_ip^i}\right)=\sum_{n=0}^{m}\left(\prod_{i=0}^{k}\binom{m_i}{n_i}\right)X^n \pmod{p}, \end{align}$

where in the final product, n_i is the ith digit in the base p representation of n. This proves Lucas's theorem.

Variations and generalizations

• 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. (This result is known as Kummer's theorem.)
• Andrew Granville has given a generalization of Lucas's theorem to the case of p being a power of prime.

External links

• Lucas's Theorem at PlanetMath.org.
• Alternate Proof of Lucas'Theorem

• (part 1);
• (part 2);
• (part 3)