  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. The populations of gaps and
constellations across stages of Eratosthenes sieve are modeled exactly by
discrete dynamic systems. These models and their asymptotic behaviors provide
evidence on a number of open problems regarding gaps between prime numbers. For
Eratosthenes sieve we show that the analogue of Polignac's conjecture is true:
every gap $g=2k$ does occur in the sieve, and its asymptotic population
supports the estimates made in Hardy and Littlewood's Conjecture B. A stronger
form of Polignac's conjecture also holds for the sieve: for any gap $g=2k$,
every feasible constellation $g,g,\ldots,g$ occurs; these constellations
correspond to consecutive primes in arithmetic progression. The models also
provide evidence toward resolving a series of questions posed by Erd\"os and
Tur\'an.
