Two \(p^ 3\) variations of Lucas' theorem
type: Item
instance of: scholarly article
title: Two \(p^ 3\) variations of Lucas' theorem
zbMATH Open document ID: 0704.11001
DOI: 10.1016/0022-314X(90)90113-6
author: D. F. Bailey
published in: Journal of Number Theory
publication date: 1990-01-01T00:00:00Z
review text: The purpose of this paper is to give two variations of Lucas' theorem which established a method of computing binomial coefficients modulo a prime. The following lemma is the key to the main results: If p is a prime and \(p\geq 5\), then \(\left( \begin{matrix} kp\\ p\end{matrix} \right)=k (mod p^ 3).\) The main result is as follows. If k and r are non-negative integers, p is a prime, and \(p\geq 5\), then \[ \left( \begin{matrix} kp\\ rp\end{matrix} \right)=\left( \begin{matrix} k\\ r\end{matrix} \right)(mod p^ 3). \] The method used to prove the above two results is by induction.
Mathematics Subject Classification ID: 11A07
Mathematics Subject Classification ID: 11B65
Mathematics Subject Classification ID: 05A10
zbMATH DE Number: 4154517
zbMATH Keywords: congruence
zbMATH Keywords: prime powers
zbMATH Keywords: Lucas' theorem
zbMATH Keywords: binomial coefficients
MaRDI profile type: Publication
cites work: Binomial Coefficients Modulo a Prime
cites work: Applications of a Simple Counting Technique
description: scientific article; zbMATH DE number 4154517