Abstract | A prime number p is called elite, if only finitely many Fermat numbers
Fm = 2^2^m +1 are quadratic residues modulo p. In this paper we give a survey on these
primes. In particular, we show how to prove that 3, 5, 7, and 41 are elite primes. Then
we introduce a necessary and sufficient condition for a prime to be elite. This condition
was recently used to determine all elite primes up to 25 ·10^10 on computers. They can
be applied as bases in the famous Pepin’s test. Several open problems concerning elite
primes are presented as well. |