### 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 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

*Journal of Number Theory*,

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

}

*Journal of Number Theory*, vol. 128, no. 3, pp. 475-499. https://doi.org/10.1016/j.jnt.2007.06.008

**On Wolstenholme's theorem and its converse.** / Helou, Charles; Terjanian, Guy.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On Wolstenholme's theorem and its converse

AU - Helou, Charles

AU - Terjanian, Guy

PY - 2008/3/1

Y1 - 2008/3/1

N2 - For any positive integer n, let wn = ((2 n - 1; n - 1)) = frac(1, 2) ((2 n; n)). Wolstenholme proved that if p is a prime ≥5, then wp ≡ 1 (mod p3). 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.

AB - For any positive integer n, let wn = ((2 n - 1; n - 1)) = frac(1, 2) ((2 n; n)). Wolstenholme proved that if p is a prime ≥5, then wp ≡ 1 (mod p3). 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.

UR - http://www.scopus.com/inward/record.url?scp=38549118433&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38549118433&partnerID=8YFLogxK

U2 - 10.1016/j.jnt.2007.06.008

DO - 10.1016/j.jnt.2007.06.008

M3 - Article

VL - 128

SP - 475

EP - 499

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

IS - 3

ER -