On Gauss and Jacobsthal congruences
https://doi.org/10.22405/2226-8383-2025-26-4-174-182
Abstract
This paper is devoted to extending the classical Wolstenholme congruence for the central binomial coefficient (︀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 combinatorialdynamic 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.
About the Authors
Konstantin Igorevich Pimenov
Saint Petersburg State University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences
Ildar Nikolaevich Faizov
LLC “Yandex Technologies”
Russian Federation
Russian Federation
Igor Borisovich Zhukov
Saint Petersburg State University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences
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Review
For citations:
Pimenov K.I., Faizov I.N., Zhukov I.B. On Gauss and Jacobsthal congruences. Chebyshevskii Sbornik. 2025;26(4):174-182. (In Russ.) https://doi.org/10.22405/2226-8383-2025-26-4-174-182
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