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A necessary and sufficient condition for the primality of Fermat numbers

Michal Křížek, Lawrence Somer (2001)

Mathematica Bohemica

We examine primitive roots modulo the Fermat number F m = 2 2 m + 1 . We show that an odd integer n 3 is a Fermat prime if and only if the set of primitive roots modulo n is equal to the set of quadratic non-residues modulo n . This result is extended to primitive roots modulo twice a Fermat number.

A note on factorization of the Fermat numbers and their factors of the form 3 h 2 n + 1

Michal Křížek, Jan Chleboun (1994)

Mathematica Bohemica

We show that any factorization of any composite Fermat number F m = 2 2 m + 1 into two nontrivial factors can be expressed in the form F m = ( k 2 n + 1 ) ( 2 n + 1 ) for some odd k and , k 3 , 3 , and integer n m + 2 , 3 n < 2 m . We prove that the greatest common divisor of k and is 1, k + 0 m o d 2 n , m a x ( k , ) F m - 2 , and either 3 | k or 3 | , i.e., 3 h 2 m + 2 + 1 | F m for an integer h 1 . Factorizations of F m into more than two factors are investigated as well. In particular, we prove that if F m = ( k 2 n + 1 ) 2 ( 2 j + 1 ) then j = n + 1 , 3 | and 5 | .

Bounds for frequencies of residues of second-order recurrences modulo p r

Walter Carlip, Lawrence Somer (2007)

Mathematica Bohemica

The authors examine the frequency distribution of second-order recurrence sequences that are not p -regular, for an odd prime p , and apply their results to compute bounds for the frequencies of p -singular elements of p -regular second-order recurrences modulo powers of the prime p . The authors’ results have application to the p -stability of second-order recurrence sequences.

Characterization of power digraphs modulo n

Uzma Ahmad, Syed Husnine (2011)

Commentationes Mathematicae Universitatis Carolinae

A power digraph modulo n , denoted by G ( n , k ) , is a directed graph with Z n = { 0 , 1 , , n - 1 } as the set of vertices and E = { ( a , b ) : a k b ( mod n ) } as the edge set, where n and k are any positive integers. In this paper we find necessary and sufficient conditions on n and k such that the digraph G ( n , k ) has at least one isolated fixed point. We also establish necessary and sufficient conditions on n and k such that the digraph G ( n , k ) contains exactly two components. The primality of Fermat number is also discussed.

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