Sophie Germain little suns

Michal Křížek; Lawrence Somer

Displaying similar documents to “Sophie Germain little suns”

On a connection of number theory with graph theory

Lawrence Somer, Michal Křížek (2004)

Czechoslovak Mathematical Journal

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We assign to each positive integer $n$ a digraph whose set of vertices is $H = {0, 1, \dots, n - 1}$ and for which there is a directed edge from $a \in H$ to $b \in H$ if $a^{2} \equiv b (m o d n)$ . 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.

17 necessary and sufficient conditions for the primality of Fermat numbers

Michal Křížek, Lawrence Somer (2003)

Acta Mathematica et Informatica Universitatis Ostraviensis

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The structure of digraphs associated with the congruence $x^{k} \equiv y (mod n)$

Lawrence Somer, Michal Křížek (2011)

Czechoslovak Mathematical Journal

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We assign to each pair of positive integers $n$ and $k \geq 2$ a digraph $G (n, k)$ whose set of vertices is $H = {0, 1, \dots, n - 1}$ and for which there is a directed edge from $a \in H$ to $b \in H$ if $a^{k} \equiv b (mod n)$ . We investigate the structure of $G (n, k)$ . In particular, upper bounds are given for the longest cycle in $G (n, k)$ . We find subdigraphs of $G (n, k)$ , called fundamental constituents of $G (n, k)$ , for which all trees attached to cycle vertices are isomorphic.