Abstract
A prime number p is called elite if only finitely many Fermat numbers 22n + 1 are quadratic residues modulo p. So far, all 21 elite primes less than 250 billion were known, together with 24 larger items. We completed the systematic search up to the range of 2.5 · 1012 finding six more such numbers. Moreover, 42 new elites larger than this bound were found, among which the largest has 374 596 decimal digits. A survey on the knowledge about elite primes together with some open problems and conjectures are presented.
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