  We study the gaps between consecutive prime numbers directly through
Eratosthenes sieve. Using elementary methods, we identify a recursive relation
for these gaps and for specific sequences of consecutive gaps, known as
constellations. Using this recursion we can estimate the numbers of a gap or of
a constellation that occur between a prime and its square. This recursion also
has explicit implications for open questions about gaps between prime numbers,
including three questions posed by Erd\"os and Tur\'an.
