Lucas's theorem

In number theory, Lucas's theorem expresses the remainder of division of the binomial coefficient $(\genfrac{}{}{0ex}{}{m}{n})$ 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:

$(\genfrac{}{}{0ex}{}{m}{n})\equiv \prod _{i=0}^k(\genfrac{}{}{0ex}{}{m_i}{n_i})(\mathrm{mod}p),$

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 $(\genfrac{}{}{0ex}{}{m}{n})=0$ if m < n.

Proofs

There are several ways to prove Lucas's theorem.

Combinatorial proof

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 $(\genfrac{}{}{0ex}{}{m}{n})$ 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 $\prod _{i=0}^k(\genfrac{}{}{0ex}{}{m_i}{n_i})(\mathrm{mod}p)$.

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

$(\genfrac{}{}{0ex}{}{p}{n})=\frac{p\cdot (p-1)\cdots (p-n+1)}{n\cdot (n-1)\cdots 1}$

is divisible by p but the denominator is not. Hence p divides $(\genfrac{}{}{0ex}{}{p}{n})$. In terms of ordinary generating functions, this means that

$(1+X{)}^p\equiv 1+X^p(\mathrm{mod}p).$

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

$(1+X{)}^{p^i}\equiv 1+X^{p^i}(\mathrm{mod}p).$

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

$\begin{array}{rl}\sum _{n=0}^m(\genfrac{}{}{0ex}{}{m}{n})X^n& =(1+X{)}^m=\prod _{i=0}^k{((1+X{)}^{p^i})}^{m_i}\\ & \equiv \prod _{i=0}^k{(1+X^{p^i})}^{m_i}=\prod _{i=0}^k\left(\sum _{n_i=0}^{m_i}(\genfrac{}{}{0ex}{}{m_i}{n_i})X^{n_i{p}^i}\right)\\ & =\prod _{i=0}^k\left(\sum _{n_i=0}^{p-1}(\genfrac{}{}{0ex}{}{m_i}{n_i})X^{n_i{p}^i}\right)=\sum _{n=0}^m\left(\prod _{i=0}^k(\genfrac{}{}{0ex}{}{m_i}{n_i})\right)X^n(\mathrm{mod}p),\end{array}$

where 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 $(\genfrac{}{}{0ex}{}{m}{n})$ 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, $(\genfrac{}{}{0ex}{}{m}{n})$ is odd if and only if the binary digits (bits) in the binary expansion of n are a subset of the bits of m.

Variations and generalizations

• Kummer's theorem asserts that the largest integer k such that p^k divides the binomial coefficient $(\genfrac{}{}{0ex}{}{m}{n})$ (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.

• Generalizations of Lucas's theorem to the case of p being a prime power are given by Davis and Webb (1990) and Granville (1997).

• The q-Lucas theorem is a generalization for the q-binomial coefficients, first proved by J. Désarménien.

External links

• Lucas's Theorem at PlanetMath.org.
• A. Laugier; M. P. Saikia (2012). "A new proof of Lucas' Theorem". Notes on Number Theory and Discrete Mathematics 18 (4): 1–6. http://nntdm.net/papers/nntdm-18/NNTDM-18-4-01-06.pdf. 
• R. Meštrović (2014). "Lucas' theorem: its generalizations, extensions and applications (1878–2014)". arXiv:1409.3820 [math.NT].