Abstract:
A sieve is constructed for ordinary twin primes of the form 6m+/-1 that are characterized by their twin rank m. It has no parity problem. Non-rank numbers are identified and counted using odd primes p>=5. Twin- and non-ranks make up the set of positive integers. Regularities of non-ranks allow gathering information on them to obtain a Legendre-type formula for the number of twin-ranks near primorial arguments.

Abstract:
In this short paper we will show, via elementary arguments, the equivalence of the Twin Prime Conjecture to a problem which might be simpler to prove. Some conclusions are drawn, and it is shown that proving the Twin Prime Conjecture is equivalent to proving that there cannot be an infinite string of consecutive natural numbers satisfying some specified equations.

Abstract:
In this paper we use the connected sum operation on knots to show that there is a one-to-one relation between knots and numbers. In this relation prime knots are bijectively assigned with prime numbers such that the prime number 2 corresponds to the trefoil knot. From this relation we have a classification table of knots where knots are one-to-one assigned with numbers. Further this assignment for the $n$th induction step of the number $2^n$ is determined by this assignment for the previous $n-1$ steps. From this induction of assigning knots with numbers we can solve some problems in number theory such as the Goldbach Conjecture and the Twin Prime Conjecture.

Abstract:
This article is a collected information from some books and papers, and in most cases the original sentences is reserved about twin prime conjecture.

Abstract:
Let E be an elliptic curve over Q. In 1988, Koblitz conjectured a precise asymptotic for the number of primes p up to x such that the order of the group of points of E over the finite field F_p is prime. This is an analogue of the Hardy and Littlewood twin prime conjecture in the case of elliptic curves. Koblitz's conjecture is still widely open. In this paper we prove that Koblitz's conjecture is true on average over a two-parameter family of elliptic curves. One of the key ingredients in the proof is a short average distribution result in the style of Barban-Davenport-Halberstam, where the average is taken over twin primes and their differences.

Abstract:
In this note we describe weight functions that exhibit a transitional behavior between weak and strong correlation with the Liouville function. We also describe a binary problem which may be considered as an interpolation between Chowla's conjecture for two-point correlations of the M\"obius function and the twin prime conjecture, in view of recent parity breaking results of K. Matom\"aki, M. Radziwi{\l}{\l} and T. Tao.

Abstract:
For integers x and k, let T(x;2k) denote the number of twin prime pairs (p,p+2k) with a distance 2k<=2x**0.5 and p<=x (not p+2k<=x). Let Tg(x;2x**0.5) denote the average of T(x;2k) for all 2k<=2x**0.5. Logically, T(x;2k) should be a function of Tg(x;2x**0.5). We first, propose a sliding model to estimate Tg(x;2x**0.5). Second, derive the relations between T(x;2k) and Tg(x;2x**0.5) from the sieve structure. Third, settle the errors caused by the dependence of primes.

Abstract:
Jing Run Chen proved in 1966 that $p+2$ has at most two prime factors for infinitely many primes $p$. However, due to the parity problem we do not know whether $p+2$ has an odd (or even) number of prime factors infinitely often. In the present work it is proved that $p+d$ has an odd number of prime factors for at least one value of d=2,4,...16.

Abstract:
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.