  One field of particular interest in Number Theory concerns the gaps between
consecutive primes. Within the last few years, very important results have been
achieved on how small these gaps can be. The strongest of these results were
obtained by Dan Goldston, Janos Pintz and Cem Yalcin Yildirim. The present work
begins by generalizing their results so that they can be applied to related
problems in a more direct manner. Additionally, we improve the bound for $F_2$
(concerning the maximal gap in a block of three primes) obtained by the
authors' first paper with our generalization.
