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

Abstract

Motivated by Euler's observation that the polynomial $x^2+x+41$ takes on prime values for $0\le x\le 39$, 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 $x^2+x+41$.

6 pages