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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 factorization of the Fermat numbers and their factors of the form 3 h 2 n + 1

Michal Křížek, Jan Chleboun (1994)

Mathematica Bohemica

We show that any factorization of any composite Fermat number F m = 2 2 m + 1 into two nontrivial factors can be expressed in the form F m = ( k 2 n + 1 ) ( 2 n + 1 ) for some odd k and , k 3 , 3 , and integer n m + 2 , 3 n < 2 m . We prove that the greatest common divisor of k and is 1, k + 0 m o d 2 n , m a x ( k , ) F m - 2 , and either 3 | k or 3 | , i.e., 3 h 2 m + 2 + 1 | F m for an integer h 1 . Factorizations of F m into more than two factors are investigated as well. In particular, we prove that if F m = ( k 2 n + 1 ) 2 ( 2 j + 1 ) then j = n + 1 , 3 | and 5 | .

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