  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.
