Square Lehmer numbers
Wayne McDaniel (1993)
Colloquium Mathematicae
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Wayne McDaniel (1993)
Colloquium Mathematicae
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Danilov, L.I. (2006)
Sibirskie Ehlektronnye Matematicheskie Izvestiya [electronic only]
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Yasushige Watase (2014)
Formalized Mathematics
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This article provides a formalized proof of the so-called “the four-square theorem”, namely any natural number can be expressed by a sum of four squares, which was proved by Lagrange in 1770. An informal proof of the theorem can be found in the number theory literature, e.g. in [14], [1] or [23]. This theorem is item #19 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/.
J. Browkin, A. Schinzel (1995)
Colloquium Mathematicae
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W. Sierpiński asked in 1959 (see [4], pp. 200-201, cf. [2]) whether there exist infinitely many positive integers not of the form n - φ(n), where φ is the Euler function. We answer this question in the affirmative by proving Theorem. None of the numbers (k = 1, 2,...) is of the form n - φ(n).
Maohua Le (1991)
Colloquium Mathematicae
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Robertson, John P. (2009)
Acta Mathematica Academiae Paedagogicae Nyí regyháziensis. New Series [electronic only]
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Antone Costa (1992)
Acta Arithmetica
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Jan Górowski, Adam Łomnicki (2014)
Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica
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In this paper a remarkable simple proof of the Gauss’s generalization 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 integer are presented, in particular, a condition for the existence of the unique element of order 2 in such a group.
Nurakunov, A.M. (2001)
Siberian Mathematical Journal
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