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The structure of digraphs associated with the congruence \(x^k\equiv y \pmod n\). (English) Zbl 1249.11006

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

MSC:

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