On the Factorization of Numbers of the Form X^2+c

Abstract

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 (Un,Zn) generating factorizations, i.e. such that: (Un)^2+c= ZnZn+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.