Lucas's 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 ${\displaystyle {\tbinom {m}{n}}}$ 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.^[1]

Statement

For non-negative integers m and n and a prime p, the following congruence relation holds:

${\displaystyle {\binom {m}{n}}\equiv \prod _{i=0}^{k}{\binom {m_{i}}{n_{i}}}{\pmod {p}},}$

where

${\displaystyle m=m_{k}p^{k}+m_{k-1}p^{k-1}+\cdots +m_{1}p+m_{0},}$

and

${\displaystyle n=n_{k}p^{k}+n_{k-1}p^{k-1}+\cdots +n_{1}p+n_{0}}$

are the base p expansions of m and n respectively. This uses the convention that ${\displaystyle {\tbinom {m}{n}}=0}$ 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 ${\displaystyle {\tbinom {m}{n}}}$ 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 ${\displaystyle \prod _{i=0}^{k}{\binom {m_{i}}{n_{i}}}{\pmod {p}}}$.

Proof based on generating functions

This proof is due to Nathan Fine.^[2]

If p is a prime and n is an integer with 1 ≤ n ≤ p − 1, then the numerator of the binomial coefficient

${\displaystyle {\binom {p}{n}}={\frac {p\cdot (p-1)\cdots (p-n+1)}{n\cdot (n-1)\cdots 1}}}$

is divisible by p but the denominator is not. Hence p divides ${\displaystyle {\tbinom {p}{n}}}$. In terms of ordinary generating functions, this means that

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

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

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

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

${\displaystyle {\begin{aligned}\sum _{n=0}^{m}{\binom {m}{n}}X^{n}&=(1+X)^{m}=\prod _{i=0}^{k}\left((1+X)^{p^{i}}\right)^{m_{i}}\\&\equiv \prod _{i=0}^{k}\left(1+X^{p^{i}}\right)^{m_{i}}=\prod _{i=0}^{k}\left(\sum _{j_{i}=0}^{m_{i}}{\binom {m_{i}}{j_{i}}}X^{j_{i}p^{i}}\right)\\&=\prod _{i=0}^{k}\left(\sum _{j_{i}=0}^{p-1}{\binom {m_{i}}{j_{i}}}X^{j_{i}p^{i}}\right)\\&=\sum _{n=0}^{m}\left(\prod _{i=0}^{k}{\binom {m_{i}}{n_{i}}}\right)X^{n}{\pmod {p}},\end{aligned}}}$

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 ${\displaystyle {\tbinom {m}{n}}}$ 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, ${\displaystyle {\tbinom {m}{n}}}$ 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 ${\displaystyle {\tbinom {m}{n}}}$ 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.

${\displaystyle {\binom {pa+r}{pb+s}}\equiv {\binom {a}{b}}{\binom {r}{s}}(1+pa(H_{r}-H_{r-s})+pb(H_{r-s}-H_{s})){\pmod {p^{2}}},}$

where ${\displaystyle H_{n}=1+{\tfrac {1}{2}}+{\tfrac {1}{3}}+\cdots +{\tfrac {1}{n}}}$ is the n^th harmonic number.^[3]

Generalizations of Lucas's theorem for higher prime powers p^k are also given by Davis and Webb (1990)^[4] and Granville (1997).^[5]

Variations and generalizations

• Kummer's theorem asserts that the largest integer k such that p^k divides the binomial coefficient ${\displaystyle {\tbinom {m}{n}}}$ (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.^[6]