On the density of primes of the form $X^2+c$

Abstract

We present a method for finding large fixed-size primes of the form $X^2+c$. We study the density of primes on the sets $E_c = \{N(X,c)=X^2+c,\ X \in (2\mathbb{Z}+(c-1))\}$, $c \in \mathbb{N}^*$. We describe an algorithm for generating values of $c$ such that a given prime $p$ is the minimum of the union of prime divisors of all elements in $E_c$. We also present quadratic forms generating divisors of Ec and study the prime divisors of its terms. This paper uses the results of Dirichlet's arithmetic progression theorem [1] and the article [6] to rewrite a conjecture of Shanks [2] on the density of primes in $E_c$. Finally, based on these results, we discuss the heuristics of large primes occurrences in the research set of our algorithm.