Bernoulli Numbers, Wolstenholme's Theorem, and p^5 Variations of Lucas' Theorem

Abstract

In this note we shall improve some congruences of D.F. Bailey [Two p^3 variations of Lucas' Theorem, JNT 35(1990), pp. 208-215] to higher prime power moduli, by studying the relation between irregular pairs of the form (p,p-3) and refined version of Wolstenholme's theorem.

arXiv comment7 pages. Final version accepted by J. of Number Theory

Cited by in corpus (19)

• Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012)
• Lucas' theorem: its generalizations, extensions and applications (1878--2014)
• A congruence involving harmonic sums modulo $p^{\alpha }{q}^{\beta }$
• Multiple Harmonic Series I: Generalizations of Wolstenholme's Theorem
• A Curious Congruence Involving Alternating Harmonic Sums
• A Simple Number Theoretic Result
• Congruences Involving Multiple Harmonic Sums and Finite Multiple Zeta Values
• Congruences related to Miki's identity
• New Congruences on Multiple Harmonic Sums and Bernoulli Numbers
• A Family of Supercongruences Involving Multiple Harmonic Sums
• Lucas' theorem modulo $p^2$
• The MHS algebra and supercongruences
• Super-congruences involving trininomial coefficients
• The congruence of Wolstenholme and generalized binomial coefficients related to Lucas sequences
• Généralisation des congruences de Wolstenholme et de Morley
• The universal Kummer congruences
• Wolstenholme's theorem over Gaussian integers
• Two supercongruences related to multiple harmonic sums
• Some Families of Super Congruences Involving Alternating Multiple Harmonic Sums