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].

In this paper, we try to study the relations between Goldbach Conjecture and
the least prime number in an arithmetic progression. We give a new weakened
form of Goldbach Conjecture. We prove that this weakened form and a weakened
form of Chowla Hypothesis imply that every sufficiently large even integer may
be written as the sum of two distinct primes.

Based on Euclid's algorithm, we find a kind of special sequences which play
an interesting role in the study of primes. We call them W Sequences. They not
only ties up the distribution of primes in short interval but also enables us
to give new weakened forms of many classical problems in Number Theory. The
object of this paper is to provide a brief introduction and preliminary
analysis on this kind of special sequences.

The object of this paper is to generalize a theorem on the binomial
coefficient [4] to the case in an arithmetic progression. We will also give a
slightly stronger result than Langevin's [2].

In this note, we generalize an ancient Greek inequality about the sequence of
primes to the cases of arithmetic progressions even multivariable polynomials
with integral coefficients. We also refine Bouniakowsky's conjecture [16] and
Conjecture 2 in [22]. Moreover, we give two remarks on conjectures in [22]