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

TL;DR

This paper investigates the factorization of numbers of the form X^2 + c, introducing new methods and an algorithm that provide empirical evidence supporting a positive answer to Landau's 4th problem.

Contribution

It establishes a family of sequences generating factorizations of X^2 + c and develops a new factorization method and prime sieve based on these properties.

Findings

Existence of sequences generating factorizations of X^2 + c

Development of a new factorization method using prime subsets

Empirical results supporting a positive answer to Landau's 4th problem

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 $(U_n, Z_n)$ 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.