A 60,000 digit prime number of the form $x^{2} + x + 41$

TL;DR

This paper reports the discovery of a 60,000-digit prime number generated by a polynomial inspired by Euler's prime-generating quadratic, achieved through a specialized search and primality proof method.

Contribution

It presents the first explicit, large prime number of the form x^2 + x + 41 with 60,000 digits, constructed via a novel approach involving reducibility conditions.

Findings

Discovered a 60,000-digit prime of the form x^2 + x + 41.

Developed a method to find large primes using polynomial reducibility.

Validated primality with classical proof techniques.

Abstract

Motivated by Euler's observation that the polynomial $x^{2} + x + 41$ takes on prime values for $0 \leq x \leq 39$, we search for large values of $x$ for which $N = x^{2} + x + 41$ is prime. To apply classical primality proving results based on the factorization of $N-1$, we choose $x$ to have the form $g(y)$, chosen so that $g(y)^{2} + g(y) + 40$ is reducible. Our main result is an explicit, 60,000 digit prime number of the form $x^{2} + x + 41$.