Some binomial coefficient congruences
type: Item
instance of: scholarly article
title: Some binomial coefficient congruences
zbMATH Open document ID: 0732.11010
DOI: 10.1016/0893-9659(91)90043-U
author: D. F. Bailey
published in: Applied Mathematics Letters
publication date: 1991-01-01T00:00:00Z
review text: \textit{É. Lucas} [Sur les congruences des nombres Eulériens et des coefficients différentiels des fonctions trigonométriques, suivant un module premier, Bull. Soc. Math. Fr. 6, 49-54 (1878; JFM 10.0139.04)] has proved that if p is prime, \(n,r,n_ 0,r_ 0\) are nonnegative integers, \(n_ 0<p\), \(r_ 0<p\), then \(\left( \begin{matrix} np+n_ 0\\ rp+r_ 0\end{matrix} \right)\equiv \left( \begin{matrix} n\\ r\end{matrix} \right)\left( \begin{matrix} n_ 0\\ r_ 0\end{matrix} \right)(mod p)\). It is now proved that if p is prime, n,r,i are nonnegative integers, \(0<i<p\), then \[ \left( \begin{matrix} np\\ rp+i\end{matrix} \right)\equiv (r+1)\left( \begin{matrix} n\\ r+1\end{matrix} \right)\left( \begin{matrix} p\\ i\end{matrix} \right)(mod p^ 2). \] It is also shown that if \(p\geq 5\) is prime, \(i>0\), \(0\leq n\leq m\), \(0\leq k<p\), then \[ \left( \begin{matrix} mp^ 2\\ np^ 2+kp+i\end{matrix} \right)\equiv (n+1)\left( \begin{matrix} m\\ n+1\end{matrix} \right)\left( \begin{matrix} p^ 2\\ kp+i\end{matrix} \right)(mod p^ 3). \] The first proof proceeds by induction on n and the second by induction on m.
Mathematics Subject Classification ID: 11B65
Mathematics Subject Classification ID: 11A07
Mathematics Subject Classification ID: 05A10
zbMATH DE Number: 4210227
reviewed by: Betty Garrison
MaRDI profile type: Publication
cites work: Binomial Coefficients Modulo a Prime
cites work: Two \(p^ 3\) variations of Lucas' theorem
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description: scientific article; zbMATH DE number 4210227