Application of statistical methods to number theory (Clustering of Primes)

Abstract

For the purposes of this paper I will define clusters of N primes in M numbers, as the occurrence of N prime numbers in a range of M consecutive numbers. The maximum clustering K in a range of L consecutive numbers is defined as, the number of integers in the range which are not eliminated by any prime number less than (M+1)/2. If M and N can satisfy two conditions, then there are an infinite number of such clusterings (i.e. prime pairs). First, for all J, the maximum clustering K for the range (aJ+b to aJ+b+M−l) is greater than or equal to N. Second, there exists a range L such that for all I for the maximum clustering K in the range (IL+d to IL+d+L−1), the ratio of K/L is less than or equal to 1/N. Due to the complexity of the notation for the proof, I could not fully explain the basic equation in this abstract.