Some divisibility properties of binomial coefficients

JOURNAL OF NUMBER THEORY

卷:183

Article

Yaqubi, Daniel¹  Mirzavaziri, Madjid¹ 

[1] Ferdowsi Univ Mashhad, Dept Pure Math, POB 1159, Mashhad 91775, Iran

关键词: Binomial coefficients;    Lucas' theorem;    Euler's totient theorem;    Bernoulli numbers;    p-adic valuation;   

DOI  :  10.1016/j.jnt.2017.08.005

来源: Elsevier

【 摘 要 】

In this paper, we aim to give full or partial proofs for the following three conjectures of V. J. W . Guo and C. Krattenthaler: (1) Let a > b be positive integers, alpha, beta be any integers and p be a prime satisfying gcd(p, a) = 1. Then there exist infinitely many positive integers n for which ((an+alpha)(bn+beta)) r(mod p) for all integers r; (2) For any odd prime p, there are no positive integers a > b such that ((an)(bn)) 0 (mod pn-1) for all n >= 1; (3) For any positive integer m, there exist positive integers a and b such that am > b and ((amn)(bn)) equivalent to 0 (mod an-1) for all n >= 1. (C) 2017 Published by Elsevier Inc.

【 授权许可】

Free