Journal of Integer Sequences, Vol. 18 (2015), Article 15.5.4

The Congruence of Wolstenholme for Generalized Binomial Coefficients Related to Lucas Sequences

Christian Ballot
Département de Mathématiques et Mécanique
Université de Caen
F14032 Caen Cedex


In recent years much research has been carried out on extending Wolstenholme classical congruence modulo the cube of a prime to higher prime powers. Here we show that this work can be done in much broader generality by replacing ordinary binomials by Lucasnomials, which are generalized binomial coefficients related to fundamental Lucas sequences. The paper builds on earlier work of Kimball and Webb in relation to the Fibonacci sequence and on recent work of the author related to congruences involving sums of quotients of Lucas sequences. The paper offers what may be a surprising line of development for very classical congruences.

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Received October 24 2014; revised versions received December 18 2014; February 25 2015; March 21 2015; March 26 2015. Published in Journal of Integer Sequences, May 19 2015.

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