欧拉商的同余式及其应用(Ⅲ)

A Congruence Involving the Quotients of Euler and Its Applications(Ⅲ)

在2002,2007的文章中,蔡天新等人介绍了一系列关于二项式系数模平方数的同余式.本文将这些同余式进行改进并推广到了模为立方数的情形,得到了许多新的同余式.如对任意正整数k和正奇数n,当e=2,3,4和6时,Πd|n([d/e]^kd-1)^μ(n/d)模n3的同余式,以及下面这类有趣的同余式Πd|n((「d/e」kd-1)/2(kd-1)/2)^^μ(n/d)=2^-(k-1)Ф(n){(modn^3/3)若3|n,(modn^3),若3|n.

In the papers of 2002 and 2007, Cai et al. introduced a series of congruences involving binomial coefficients under perfect moduli. This article generalizes these congruences to cubic cases leading to many new statements. For example, the congruence Πd|n([d/e]kd-1)^μ(n/d) module n3 for e=2, 3, 4 and 6, and the following congruence Πd|n((d-1)/2(kd-1)/2)^^μ(n/d)=2^-(k-1)Ф(n){(modn^3/3)if3|n,(modn^3),if3|n.