## Experimental Mathematics

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

### Searching for Large Elite Primes

#### 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

**Permanent link to this document**

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