The Most Likely Common Difference Of Arithmetic Progressions Among Primes

The key to understanding the prime number theory is to study the distribution of primes,and the difference between primes is the core content that reveals the distribution of primes.The arithmetic progressions composed of primes is called prime arithmetic progressions.In order to further understand the distribution of primes,this paper mainly discusses the common difference distribution of arithmetic progressions of primes.It is widely known by Green-Tao theorem that there are arbitrarily long arithmetic progressions among primes,however the distribution of common differences of these arithmetic progressions is still mysterious for us.For any sufficiently large x,select k+1 primes below x to form prime arithmetic progression n,n+d,n+2d,…,n+kd with common difference d.Let dk*(x)be the most likely common differences of all arithmetic progressions.Based on the truth of Hardy-Littlewood Conjecture,we obtain that(?)uniformly in k,and every prime divides all sufficiently large most likely common differences.