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.

翻译：我们提出了一种寻找形如$X^2+c$的大尺寸固定素数的方法。我们研究集合$E_c = \{N(X,c)=X^2+c,\ X \in (2\mathbb{Z}+(c-1))\}$（其中$c \in \mathbb{N}^*$）上的素数密度。我们描述了一种生成$c$值的算法，使得给定素数$p$是$E_c$中所有元素素因子并集的最小值。我们还提出了生成$E_c$因子的二次型，并研究其各项的素因子。本文利用狄利克雷算术级数定理[1]的结果以及文献[6]，重新表述了Shanks[2]关于$E_c$中素数密度的猜想。最后，基于这些结果，我们讨论了算法研究集中大素数出现的启发式方法。