Eratosthenes sieve and the gaps between primes
  A few years ago we identified a recursion that works directly with the gaps among the generators in each stage of Eratosthenes sieve. This recursion provides explicit enumerations of sequences of gaps among the generators, which sequences are known as constellations.   By studying this recursion on the cycles of gaps across stages of Eratosthenes sieve, we are able to provide evidence on a number of open problems regarding gaps between prime numbers. The basic counts of short constellations in the cycles of gaps provide evidence toward the twin prime conjecture and toward resolving a series of questions posed by Erdos and Turan. The dynamic system underlying the recursion provides evidence toward Polignac's conjecture and in support of the estimates made for gaps among primes by Hardy and Littlewood in Conjecture B of their 1923 paper. 
Fred B. Holt and Helgi Rudd
math.NT