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1. A necessary and sufficient condition for the primality of Fermat numbers

Creator:
Křížek, Michal and Somer, Lawrence
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
Fermat numbers, primitive roots, primality, and Sophie Germain primes
Language:
English
Description:
We examine primitive roots modulo the Fermat number Fm = 2 2m + 1. We show that an odd integer n ≥ 3 is a Fermat prime if and only if the set of primitive roots modulo n is equal to the set of quadratic non-residues modulo n. This result is extended to primitive roots modulo twice a Fermat number.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

2. On a connection of number theory with graph theory

Creator:
Somer, Lawrence and Křížek, Michal
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
Fermat numbers, Chinese remainder theorem, primality, group theory, and digraphs
Language:
English
Description:
We assign to each positive integer $n$ a digraph whose set of vertices is $H=\lbrace 0,1,\dots ,n-1\rbrace $ and for which there is a directed edge from $a\in H$ to $b\in H$ if $a^2\equiv b\hspace{4.44443pt}(\@mod \; n)$. 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.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public