Open Access
2006 Searching for Large Elite Primes
Tom Müller
Experiment. Math. 15(2): 183-186 (2006).


A prime number $p$ is called elite if only finitely many Fermat numbers $2^{2^n}+1$ 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 $10^9$ 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.


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Tom Müller. "Searching for Large Elite Primes." Experiment. Math. 15 (2) 183 - 186, 2006.


Published: 2006
First available in Project Euclid: 5 April 2007

zbMATH: 1132.11005
MathSciNet: MR2253004

Primary: 11A15 , 11A41

Keywords: Elite primes , Fermat numbers

Rights: Copyright © 2006 A K Peters, Ltd.

Vol.15 • No. 2 • 2006
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