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Paulo Ribenboim (2003)

Bollettino dell'Unione Matematica Italiana

A necessary and sufficient condition for the primality of Fermat numbers

Michal Křížek, Lawrence Somer (2001)

Mathematica Bohemica

We examine primitive roots modulo the Fermat number F m = 2 2 m + 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.

Automaticity IV : sequences, sets, and diversity

Jeffrey Shallit (1996)

Journal de théorie des nombres de Bordeaux

This paper studies the descriptional complexity of (i) sequences over a finite alphabet ; and (ii) subsets of N (the natural numbers). If ( s ( i ) ) i 0 is a sequence over a finite alphabet Δ , then we define the k -automaticity of s , A s k ( n ) , to be the smallest possible number of states in any deterministic finite automaton that, for all i with 0 i n , takes i expressed in base k as input and computes s ( i ) . We give examples of sequences that have high automaticity in all bases k ; for example, we show that the characteristic...

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