Searching for Large Elite Primes

Abstract

A prime number p is called elite if only finitely many Fermat numbers 2²ⁿ + 1 are quadratic residues modulo p. Previously, only fourteen elite primes were known explicitly, all of them smaller than 35 million. Using computers, we searched all primes less than 10⁹ for other elite primes and discovered p = 159 318 017 and p = 446 960 641 as the fifteenth and sixteenth elite primes. Moreover, with another approach we found 26 other elite primes larger than a billion, the largest of which has 1172 decimal digits. Finally, we derive some conjectures about elite primes from the results of our computations.

Keywords:

• 11A15, 11A41
• elite primes
• Fermat numbers