Dickson's conjecture on $Z^n$--An equivalent form of Green-Tao's conjecture

TL;DR

This paper extends Dickson's conjecture to $Z^n$, providing an equivalent form of Green-Tao's conjecture, and suggests a general theory for simultaneous prime representation by multivariable polynomials on $Z^n$.

Contribution

It introduces a new form of Dickson's conjecture on $Z^n$ and establishes an equivalent form of Green-Tao's conjecture, advancing the understanding of prime representations in multiple variables.

Findings

Derived an equivalent form of Green-Tao's conjecture.

Proposed a general theory for multivariable polynomial prime representation.

Suggested generalization of Chinese Remainder Theorem analogy.

Abstract

In [1], we give Dickson's conjecture on $N^n$. In this paper, we further give Dickson's conjecture on $Z^n$ and obtain an equivalent form of Green-Tao's conjecture [2]. Based on our work, it is possible to establish a general theory that several multivariable integral polynomials on $Z^n$ represent simultaneously prime numbers for infinitely many integral points and generalize the analogy of Chinese Remainder Theorem in [3]. Dans [1], nous donnons la conjecture de Dickson sur $N^n$. Dans ce document, en outre nous accordons une conjecture de Dickson sur $Z^n$ et obtenons une forme \'{e}quivalent de conjecture de Green-Tao [2]. Sur la base de nos travaux, il est possible d'\'{e}tablir une th\'{e}orie g\'{e}n\'{e}rale que plusieurs polyn\^{o}mes int\'{e}graux multivariables sur $Z^n$ repr\'{e}sentent simultan\'{e}ment les nombres premiers pour un nombre infini de points entiers et de g\'{e}n\'{e}raliser les l'analogie de Th\'{e}or\`{e}me des Restes Chinois dans [3].