  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.
