### Abstract

For any positive integer n, let w_{n} = ((2 n - 1; n - 1)) = frac(1, 2) ((2 n; n)). Wolstenholme proved that if p is a prime ≥5, then w_{p} ≡ 1 (mod p^{3}). 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 w_{n}, 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.

Original language | English (US) |
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Pages (from-to) | 475-499 |

Number of pages | 25 |

Journal | Journal of Number Theory |

Volume | 128 |

Issue number | 3 |

DOIs | |

State | Published - Mar 1 2008 |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

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## Cite this

Helou, C., & Terjanian, G. (2008). On Wolstenholme's theorem and its converse.

*Journal of Number Theory*,*128*(3), 475-499. https://doi.org/10.1016/j.jnt.2007.06.008