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On a connection of number theory with graph theory. (English) Zbl 1080.11004

Summary: 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 \(a^2\equiv b\pmod n\). 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.

MSC:

11A07 Congruences; primitive roots; residue systems
05C20 Directed graphs (digraphs), tournaments
11A51 Factorization; primality
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References:

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