The Prime Glossary: gaps between primes

gaps between primes
(another Prime Pages' Glossary entries)

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Look at the first few primes: 2, 3, 5, 7, 11, and 13. Notice they have irregular gaps between them: 2 is followed immediately by the prime 3, 3 is followed by one composite, 5 by one, but 7 by three. We call the number of composites following a prime the length of the prime gap. For example, the prime gaps after 2, 3, 5 and seven are 0, 1, 1 and 3 respectively. By the prime number theorem, the "average gap" between primes less than n is log(n). See the page on prime gaps (linked below) for much more information.

Warning: Some authors define the prime gap to be the difference between consecutive primes, this is a number one larger than our definition.

Related pages (outside of this work)

The gaps between primes (an expanded article on prime gaps)
An extensive table of prime gaps by T. Nicely (updated very regularly)
Gaps between consecutive primes by Tomás Oliveira e Silva (nice graphs!)

References:

Brent74
R. P. Brent, "The distribution of small gaps between succesive primes," Math. Comp., 28 (1974) 315--324. MR 48:8356
Nicely99
T. Nicely, "New maximal prime gaps and first occurrences," Math. Comp., 68:227 (July 1999) 1311--1315. MR 99i:11004 (Abstract available) [Reprint available at http://www.trnicely.net/index.html]
NN99
T. Nicely and B. Nyman, "First occurrence of a prime gap of 1000 or greater," preprint available at http://www.trnicely.net/index.html.
YP89
J. Young and A. Potler, "First occurrence prime gaps," Math. Comp., 53:185 (1989) 221--224. MR 89f:11019 [Lists gaps between primes up to the 777 composites following 42842283925351.]