2009/06/21 by Shaohua Zhang, Zhang, Shaohua
Mathematics · #11A41 #11A99 #Analytic Number Theory Research #FOS: Mathematics #Finite Group Theory Research #General Mathematics (math.GM) #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.0906.3850
openalex publication_date 2009/06/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Nk . 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.