  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: $(U_n)^2+c = Z_nZ_{n+1}$. 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.
