## Experimental Mathematics

### Searching for Large Elite Primes

Tom Müller

#### Abstract

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.

#### Article information

Source
Experiment. Math., Volume 15, Issue 2 (2006), 183-186.

Dates
First available in Project Euclid: 5 April 2007

https://projecteuclid.org/euclid.em/1175789738

Mathematical Reviews number (MathSciNet)
MR2253004

Zentralblatt MATH identifier
1132.11005

Subjects
Primary: 11A15: Power residues, reciprocity 11A41: Primes

Keywords
Elite primes Fermat numbers

#### Citation

Müller, Tom. Searching for Large Elite Primes. Experiment. Math. 15 (2006), no. 2, 183--186. https://projecteuclid.org/euclid.em/1175789738