Notes on Dickson's Conjecture

TL;DR

This paper explores a multivariable generalization of Dickson's conjecture, providing an equivalent formulation and discussing related conjectures, with some evidence and remarks on the topic.

Contribution

It formulates an equivalent form of Dickson's conjecture and extends it to multivariable systems of affine-linear forms, filling a gap in existing generalizations.

Findings

Proposes an equivalent form of Dickson's conjecture.

Generalizes the conjecture to multivariable affine-linear systems.

Provides remarks and evidence supporting the conjecture.

Abstract

In 1904, Dickson [5] stated a very important conjecture. Now people call it Dickson's conjecture. In 1958, Schinzel and Sierpinski [14] generalized Dickson's conjecture to the higher order integral polynomial case. However, they did not generalize Dickson's conjecture to the multivariable case. In 2006, Green and Tao [13] considered Dickson's conjecture in the multivariable case and gave directly a generalized Hardy-Littlewood estimation. But, the precise Dickson's conjecture in the multivariable case does not seem to have been formulated. In this paper, based on the idea in [15], we will try to complement this and give an equivalent form of Dickson's Conjecture, furthermore, generalize it to the multivariable case or a system of affine-linear forms on $N^k$ . We also give some remarks and evidences on conjectures in [15]. Finally, in Appendix, we briefly introduce the basic theory that several multivariable integral polynomials represent simultaneously prime numbers for infinitely many integral points.