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On a connection of number theory with graph theory

Lawrence Somer, Michal Křížek (2004)

Czechoslovak Mathematical Journal

We assign to each positive integer n a digraph whose set of vertices is H = { 0 , 1 , , n - 1 } and for which there is a directed edge from a H to b H if a 2 b ( m o d n ) . We establish necessary and sufficient conditions for the existence of isolated fixed points. We also examine when the digraph is semiregular. Moreover, we present simple conditions for the number of components and length of cycles. Two new necessary and sufficient conditions for the compositeness of Fermat numbers are also introduced.

On Alternatives of Polynomial Congruences

Mariusz Skałba (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

What should be assumed about the integral polynomials f ( x ) , . . . , f k ( x ) in order that the solvability of the congruence f ( x ) f ( x ) f k ( x ) 0 ( m o d p ) for sufficiently large primes p implies the solvability of the equation f ( x ) f ( x ) f k ( x ) = 0 in integers x? We provide some explicit characterizations for the cases when f j ( x ) are binomials or have cyclic splitting fields.

On semiregular digraphs of the congruence x k y ( mod n )

Lawrence Somer, Michal Křížek (2007)

Commentationes Mathematicae Universitatis Carolinae

We assign to each pair of positive integers n and k 2 a digraph G ( n , k ) whose set of vertices is H = { 0 , 1 , , n - 1 } and for which there is a directed edge from a H to b H if a k b ( mod n ) . The digraph G ( n , k ) is semiregular if there exists a positive integer d such that each vertex of the digraph has indegree d or 0. Generalizing earlier results of the authors for the case in which k = 2 , we characterize all semiregular digraphs G ( n , k ) when k 2 is arbitrary.

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