On the factorization of numbers of the form $X^2+c$

We study the factorization of the numbers $N=X^2+c$, where $c$ is a fixed constant, and this independently of the value of gcd$(X,c)$. We prove the existence of a family of sequences with arithmetic difference $(U_n,Z_n)$ generating factorizations, i.e. such that: $(U_n{)}^2+c=Z_n{Z}_{n+1}$. The different properties demonstrated allow us to establish new factorization methods by a subset of prime numbers and to define a prime sieve. An algorithm is presented on this basis and leads to empirical results which suggest a positive answer to Landau’s 4th problem.