On Wolstenholme's theorem and its converse |
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Authors: | Charles Helou Guy Terjanian |
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Affiliation: | a Pennsylvania State University, 25 Yearsley Mill Road, Media, PA 19063, USA b Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France |
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Abstract: | For any positive integer n, let . Wolstenholme proved that if p is a prime ?5, then . The converse of Wolstenholme's theorem, which has been conjectured to be true, remains an open problem. In this article, we establish several relations and congruences satisfied by the numbers wn, and we deduce that this converse holds for many infinite families of composite integers n. In passing, we obtain a number of congruences satisfied by certain classes of binomial coefficients, and involving the Bernoulli numbers. |
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Keywords: | 11A07 11A41 11B65 11B68 |
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