  This article consists of three chapters.In Chapter 1, it is determined by the
consecutive odd numbers, and study to the intrinsic properties of a class of
matrix sequence. Through the establishment of matrix online number concept,
characteristics and the online number column use mathematical induction to
prove the some properties of this kind of matrix on the number of online
features (Theorem 1). Finally, it is given a trial to prove the Goldbach
conjecture (Theorem 6). This is the author in the years to explore prime
properties in the process of research and discovery, and believe that this
finding is of great significance.In Chapter 2, it is defined the concepts of
matrix master characteristic number and the Matrix Master Characteristic
Sequence (Definition 1). Firstly, we prove that any even number can be
expressed as for the difference of two odd prime numbers at least two groups
(Theorem 4). Secondly, we prove that there are infinitely many odd prime
numbers separated by four (Theorem 9). Finally, we prove that if there is
greater than 1 in the intersection by S(3) and s(2m+3) for any natural number
m, so that there are infinitely many odd prime numbers separated by
2m(Theorem11). The results will undoubtedly promote the research for Polignac
conjecture.In Chapter 3, mainly as a result of any odd natural number a, the
intersection by S(a) and s(a+2) is not empty number set, and there are far more
than 1 number in the set, where S(a)={k,If 2k+a be prime as k be natural
number},and P is a prime number set, N is natural number set. we prove that
there are an infinite number of twin prime, and then solve the problem of the
twin primes in number theory.
