On Wolstenholme's theorem and its converse

Charles Helou, Guy Terjanian

Research output: Contribution to journalArticle

11 Citations (Scopus)

Abstract

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.

Original language English (US) 475-499 25 Journal of Number Theory 128 3 https://doi.org/10.1016/j.jnt.2007.06.008 Published - Mar 1 2008

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Converse
Congruence
Bernoulli numbers
Integer
Binomial coefficient
Theorem
Deduce
Open Problems
Composite
Class
Family

All Science Journal Classification (ASJC) codes

• Algebra and Number Theory

Cite this

Helou, Charles ; Terjanian, Guy. / On Wolstenholme's theorem and its converse. In: Journal of Number Theory. 2008 ; Vol. 128, No. 3. pp. 475-499.
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On Wolstenholme's theorem and its converse. / Helou, Charles; Terjanian, Guy.

In: Journal of Number Theory, Vol. 128, No. 3, 01.03.2008, p. 475-499.

Research output: Contribution to journalArticle

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