## Characterization of power digraphs modulo $$n$$.(English)Zbl 1249.11002

The authors assign to each pair of positive integers $$n$$ and $$k\geq 2$$ a digraph $$G(n,k)$$ whose set of vertices is $$Z_{n}=\{0,1,\dots n-1\}$$ and for which there exists a directed edge from $$a\in Z_{n}$$ to $$b\in Z_{n}$$ if $$a^{k}\equiv b \pmod n$$. They establish necessary and sufficient conditions such that $$G(n,k)$$ has at least one isolated fixed point. Another necessary and sufficient condition on $$n$$ and $$k$$, such that the digraph $$G(n,k)$$ contains exactly two components, is given. A new necessary and sufficient condition for the primality of the Fermat numbers is presented as well.

### MSC:

 11A07 Congruences; primitive roots; residue systems 11A15 Power residues, reciprocity 05C20 Directed graphs (digraphs), tournaments 11A51 Factorization; primality
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