Square Lehmer numbers
Wayne McDaniel (1993)
Colloquium Mathematicae
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Wayne McDaniel (1993)
Colloquium Mathematicae
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Antone Costa (1992)
Acta Arithmetica
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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).
Robertson, John P. (2009)
Acta Mathematica Academiae Paedagogicae Nyí regyháziensis. New Series [electronic only]
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J. H. E. Cohn (1992)
Acta Arithmetica
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Nurakunov, A.M. (2001)
Siberian Mathematical Journal
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Danilov, L.I. (2006)
Sibirskie Ehlektronnye Matematicheskie Izvestiya [electronic only]
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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.
Victor Abrashkin (2010)
Journal de Théorie des Nombres de Bordeaux
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A local analogue of the Grothendieck Conjecture is an equivalence between the category of complete discrete valuation fields with finite residue fields of characteristic and the category of absolute Galois groups of fields together with their ramification filtrations. The case of characteristic 0 fields was studied by Mochizuki several years ago. Then the author of this paper proved it by a different method in the case (but with no restrictions on the characteristic of )....
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/.