On The Distribution Of Arithmetic Progressions In Three Prime Numbers

Arithmetic progressions in the primes is a problem of great concern in Number Theory.The famous Green and Tao in mathematics proved that there are arbitrarily long arithmetic progressions in primes,but we still can't get much information about the common difference.Let Dk={d₁,d₂…d_k} be a set of k distinct integers with d₁<d₂<…<d_k and ?(x,Dk)denote the number of positive integers n?x such that n+d₁,…,n+d_k.In 1923,Hardy and Littlewood conjectured an asymptotic formula for ?(x,Dk).This thesis is based on the conjecture,trying to further study the distribution information of the common difference of arithmetic progressions of primes of length three.Assuming an appropriate form of Hardy-Littlewood Conjecture,we get the following conclusions:On the one hand,the most frequently occurring difference of arithmetic progressions of primes of length three tends to infinity.And for any sufficiently large x,the most frequently occurring difference are primorials.On the other hand,we present two examples to indicate that the arithmetic progressions is not necessarily the most frequently occurring difference of three prime numbers.