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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.

A note on the congruence n p k m p k n m ( mod p r )

Romeo Meštrović (2012)

Czechoslovak Mathematical Journal

In the paper we discuss the following type congruences: n p k m p k m n ( mod p r ) , where p is a prime, n , m , k and r are various positive integers with n m 1 , k 1 and r 1 . Given positive integers k and r , denote by W ( k , r ) the set of all primes p such that the above congruence holds for every pair of integers n m 1 . Using Ljunggren’s and Jacobsthal’s type congruences, we establish several characterizations of sets W ( k , r ) and inclusion relations between them for various values k and r . In particular, we prove that W ( k + i , r ) = W ( k - 1 , r ) for all k 2 , i 0 and 3 r 3 k , and W ( k , r ) = W ( 1 , r ) for...

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