Simple proofs of some generalizations of the Wilson’s theorem

Jan Górowski; Adam Łomnicki

Simple proofs of some generalizations of the Wilson’s theorem

Authors

Jan Górowski
Adam Łomnicki

Abstract

In this paper a remarkable simple proof of the Gauss’s generaliza tion of the Wilson’s theorem is given. The proof is based on properties of a subgroup generated by element of order 2 of a finite abelian group. Some conditions equivalent to the cyclicity of (Φ(n), ·n ), where n > 2 is an inte ger are presented, in particular, a condition for the existence of the unique element of order 2 in such a group.

References

Lin Cong, Zhipeng Li, On Wilson’s theorem and Polignac conjecture, Math. Medley 32 (2005), 11–16. (arXiv:math/0408018v1).
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J.B. Cosgrave, K. Dilcher, Extensions of the Gauss-Wilson theorem, Integers 8 (2008), A39, 15pp.
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M. Hassani, M. Momeni-Pour, Euler type generalization of Wilson’s theorem, arXiv:math/0605705v1 28 May, 2006.
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G.A. Miller, A new proof of the generalized Wilson’s theorem, Ann. of Math. (2) 4 (1903), 188–190.
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Published

2014-12-11

How to Cite

Górowski, J., & Łomnicki, A. (2014). Simple proofs of some generalizations of the Wilson’s theorem. Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica, 13, 7–14. Retrieved from https://studmath.uken.krakow.pl/article/view/13193