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

M. Wolf, Franccois 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 (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.
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关于形式为X^2+c的数的因数分解
我们研究了N=X^2+c的因式分解,其中c是一个固定常数,并且这与gcd的值无关。证明了一组算术差(Un,Zn)产生因数分解的序列的存在性,即:(Un)^2+c= ZnZn+1。所证明的不同性质使我们能够通过质数子集建立新的因数分解方法并定义质数筛。在此基础上提出了一种算法,并得出了实证结果,对朗道第四问题给出了积极的答案。
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