A q-analog of Ljunggren's binomial congruence

2011/03/16 by Armin Straub, Straub, Armin · 4 citations
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #math.CO #math.NT

paper · pdf · doi:10.48550/arxiv.1103.3258

6 pages, to be published in the proceedings of FPSAC 2011

arxiv created 2011/03/16 · arxiv updated 2011/03/17

Abstract

We prove a q-analog of a classical binomial congruence due to Ljunggren which states that \binoma pb p ≡ \binomab modulo p³ for primes p≥5. This congruence subsumes and builds on earlier congruences by Babbage, Wolstenholme and Glaisher for which we recall existing q-analogs. Our congruence generalizes an earlier result of Clark.

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• Some q-analogues of (super)congruences of Beukers, Van Hamme and Rodriguez-Villegas
• Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012)
• Ljunggren--Jacobsthal and Bailey-Type Congruences for Rectangular Gaussian Binomial Coefficients
• Notes on q-analogues of a binomial congruence of Glaisher

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