For any positive integer n, let w n = 2n−1 n−1 = 1 2 2n 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.

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Sur une question de V.A. Lebesgue Annales de l'institut Fourier, tome 37, n o 3 (1987), p. 19-37 © Annales de l'institut Fourier, 1987, tous droits réservés. L'accès aux archives de la revue « Annales de l'institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

Wendt's determinant of order n is the circulant determinant W n whose (i, j )-th entry is the binomial coefficient n |i−j | , for 1 i, j n, where n is a positive integer. We establish some congruence relations satisfied by these rational integers. Thus, if p is a prime number and k a positive integer, then W p k ≡ 1 (mod p k ) and W np k ≡ W n (mod p). If q is another prime, distinct from p, and h any positive integer, then W p h q k ≡ W p h W q k (mod pq). Furthermore, if p is odd,In particular, if p 5, then W p ≡ 1 (mod p 4 ). Also, if m and n are relatively prime positive integers, then W m W n divides W mn .