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.

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

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

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

where in the final product, n_i is the ith digit in the base p representation of n. This proves 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 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.
• Andrew Granville has given a generalization of Lucas's theorem to the case of p being a power of prime.[3]

• The q-Lucas theorem is a generalization for the q-binomial coefficients, first proved by J. Désarménien.[4]

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.
4. Désarménien, Jacques (March 1982). "Un Analogue des Congruences de Kummer pour les q-nombres d'Euler". European Journal of Combinatorics. 3 (1): 19–28. doi:10.1016/S0195-6698(82)80005-X.

External links

• Lucas's Theorem at PlanetMath.org.
• A. Laugier; M. P. Saikia (2012). "A new proof of Lucas' Theorem" (PDF). Notes on Number Theory and Discrete Mathematics. 18 (4): 1–6. arXiv:1301.4250.
• R. Meštrović (2014). "Lucas' theorem: its generalizations, extensions and applications (1878--2014)". Preprint. arXiv:/1409.3820.