On Gauss and Jacobsthal congruences

K. I. Pimenov^a, I. N. Faizov^b, I. B. Zhukov<sup>a

a</sup> Saint Petersburg State University (St. Petersburg)
^b LLC “Yandex Technologies” (St. Petersburg)

Abstract: This paper is devoted to extending the classical Wolstenholme congruence for the central binomial coefficient $\binom{2p}{p}$ to the case of a composite number. An extension of Fermat's little theorem to the composite case is the Gauss congruence, which has a simple combinatorial-dynamic interpretation. To extend Wolstenholme's congruence to the composite case, it is necessary to use the Jacobsthal congruence. A combinatorial proof of its weakened version is given based on investigation of the orbit lenghts for a suitable action of Sylow $p$-subgroups of the symmetric group.

Keywords: Fermat little theorem, elementary number theory, arithmetical dynamics, Sylow subgroup, Gauss congruence, Dold Sequence,Wolstenholme's theorem.

UDC: 511.17

Received: 13.06.2025
Accepted: 17.10.2025

DOI: 10.22405/2226-8383-2025-26-4-174-182