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arXiv · 2208.12579

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

Marc Wolf·François Wolf
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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.

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Marc Wolf, François Wolf. 2022-07-23. On the factorization of numbers of the form $X^2+c$. https://arxiv.org/abs/2208.12579

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