17 necessary and sufficient conditions for the primality of Fermat numbers
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Michal Křížek, Lawrence Somer (2003)
Acta Mathematica et Informatica Universitatis Ostraviensis
Kilford, L.J.P. (2004)
International Journal of Mathematics and Mathematical Sciences
Pedro Berrizbeitia, T. G. Berry, Juan Tena-Ayuso (2003)
Acta Arithmetica
Michal Křížek, Lawrence Somer (2001)
Mathematica Bohemica
We examine primitive roots modulo the Fermat number . We show that an odd integer is a Fermat prime if and only if the set of primitive roots modulo is equal to the set of quadratic non-residues modulo . This result is extended to primitive roots modulo twice a Fermat number.
Vsemirnov, M. (2004)
Journal of Integer Sequences [electronic only]
Michal Křížek, Jan Chleboun (1994)
Mathematica Bohemica
We show that any factorization of any composite Fermat number into two nontrivial factors can be expressed in the form for some odd and , and integer . We prove that the greatest common divisor of and is 1, , and either or , i.e., for an integer . Factorizations of into more than two factors are investigated as well. In particular, we prove that if then and .
František Marko (1996)
Mathematica Slovaca
Stephen S.-T. Yau, Yi-Jing Xu (1995)
Journal für die reine und angewandte Mathematik
Vladimír Puš (1989)
Commentationes Mathematicae Universitatis Carolinae
Jones, Lenny, White, Daniel (2011)
Journal of Integer Sequences [electronic only]
Walter Carlip, Lawrence Somer (2007)
Mathematica Bohemica
The authors examine the frequency distribution of second-order recurrence sequences that are not -regular, for an odd prime , and apply their results to compute bounds for the frequencies of -singular elements of -regular second-order recurrences modulo powers of the prime . The authors’ results have application to the -stability of second-order recurrence sequences.
Jean-Marie De Koninck, Nicolas Doyon, A. Arthur Bonkli Razafindrasoanaivolala, William Verreault (2020)
Archivum Mathematicum
A positive integer is said to be a Jordan-Pólya number if it can be written as a product of factorials. We obtain non-trivial lower and upper bounds for the number of Jordan-Pólya numbers not exceeding a given number .
Ladislav Skula (2003)
Δελτίο της Ελληνικής Μαθηματικής Εταιρίας
Uzma Ahmad, Syed Husnine (2011)
Commentationes Mathematicae Universitatis Carolinae
A power digraph modulo , denoted by , is a directed graph with as the set of vertices and as the edge set, where and are any positive integers. In this paper we find necessary and sufficient conditions on and such that the digraph has at least one isolated fixed point. We also establish necessary and sufficient conditions on and such that the digraph contains exactly two components. The primality of Fermat number is also discussed.
Hamahata, Y., Kokubun, Y. (2007)
Journal of Integer Sequences [electronic only]
Vladimír Puš (1991)
Czechoslovak Mathematical Journal
Vladimir Shevelev (2007)
Acta Arithmetica
Florian Luca (2005)
Δελτίο της Ελληνικής Μαθηματικής Εταιρίας
H. Schapira (1980)
Jahresbericht der Deutschen Mathematiker-Vereinigung
Bernd Richter (1972)
Journal für die reine und angewandte Mathematik
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