Lucas's theorem

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] Contents 1 Statement 2 Consequences 3 Variations and generalizations 4 References 5 External links 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 mi 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 Cpi 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 ni 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 {pcdot (p-1)cdots (p-n+1)}{ncdot (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 mi with 0 ≤ mi ≤ 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, ni 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. 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. Generalizations of Lucas's theorem to the case of p being a prime power are given by Davis and Webb (1990)[3] and Granville (1997).[4] The q-Lucas theorem is a generalization for the q-binomial coefficients, first proved by J. Désarménien.[5] References ^ 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) ^ Fine, Nathan (1947). "Binomial coefficients modulo a prime". American Mathematical Monthly. 54 (10): 589–592. doi:10.2307/2304500. JSTOR 2304500. ^ Kenneth S. Davis, William A. Webb (1990). "Lucas' Theorem for Prime Powers". European Journal of Combinatorics. 11 (3): 229–233. doi:10.1016/S0195-6698(13)80122-9. ^ 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. ^ 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. 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)". arXiv:1409.3820 [math.NT]. Categories: Theorems about prime numbers

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