# Lucas's 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.

## Consequence

• 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.

## Variations and generalizations

• Kummer's theorem asserts that the largest integer k such that pk 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.
• Andrew Granville has given a generalization of Lucas's theorem to the case of p being a power of prime.[3]

## References

1. ^
• Edouard Lucas (1878). "Théorie des Fonctions Numériques Simplement Périodiques". American Journal of Mathematics. 1 (2): 184–196. doi:10.2307/2369308. JSTOR 2369308. MR 1505161. (part 1);
• Edouard Lucas (1878). "Théorie des Fonctions Numériques Simplement Périodiques". American Journal of Mathematics. 1 (3): 197–240. doi:10.2307/2369311. JSTOR 2369311. MR 1505164. (part 2);
• Edouard Lucas (1878). "Théorie des Fonctions Numériques Simplement Périodiques". American Journal of Mathematics. 1 (4): 289–321. doi:10.2307/2369373. JSTOR 2369373. MR 1505176. (part 3)
2. ^ Fine, Nathan (1947). "Binomial coefficients modulo a prime". American Mathematical Monthly. 54: 589–592. doi:10.2307/2304500.
3. ^ Andrew Granville (1997). "Arithmetic Properties of Binomial Coefficients I: Binomial coefficients modulo prime powers" (PDF). Canadian Mathematical Society Conference Proceedings. 20: 253–275. MR 1483922. Archived from the original (PDF) on 2017-02-02.