View source for Prime gaps

If <math>p_n</math> denotes the n^th prime, then <math>p_{n+1}-p_n</math> is the n^th prime gap.

On average, the prime number theorem tells us that <math>p_{n+1}-p_n</math> has size <math>O(\log p_n)</math>.  

A recent result of Goldston-Pintz-Yildirim shows that there exist infinitely many n for which the gap is as small as <math>o(\log p_n)</math> (in fact more precise bounds are known).  But the set of small gaps established by this method is sparse.

[[Cramer's conjecture]] asserts that the prime gap never exceeds <math>O(\log^2 p_n)</math> in size.  If so, this resolves the [[finding primes]] project positively.  However, the best upper bound on the prime gap is <math>O( p_n^{1/2} \log p_n )</math> assuming the Riemann hypothesis, and <math>O( p_n^{0.535} )</math> otherwise (a result of Baker and Harman).

Rankin showed that the prime gap can be as large as <math>\log p_n \frac{\log \log p_n \log \log \log \log p_n}{(\log \log \log p_n)^3}</math>.

# R. C. Baker and G. Harman, “The difference between consecutive primes,” Proc. Lond. Math. Soc., series 3, 72 (1996) 261–280. MR 96k:11111
# K. Soundararajan, [http://arxiv.org/abs/math/0605696 Small gaps between prime numbers: The work of Goldston-Pintz-Yildirim]
# [http://en.wikipedia.org/wiki/Prime_gap The Wikipedia entry on prime gaps]