Prime gaps

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If $p_n$ denotes the n^th prime, then $p_{n+1}-p_n$ is the n^th prime gap.

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

A recent result of Goldston-Pintz-Yildirim shows that there exist infinitely many n for which the gap is as small as $o(\log p_n)$ (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 $O(\log^2 p_n)$ in size. If so, this resolves the finding primes project positively. However, the best upper bound on the prime gap is $O( p_n^{1/2} \log p_n )$ assuming the Riemann hypothesis, and $O( p_n^{0.525} )$ otherwise (a result of Baker, Harman, and Pintz; an earlier bound of $O(p_n^{0.535})$ was obtained by Baker and Harman.).

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

1. R. C. Baker and G. Harman, “The difference between consecutive primes,” Proc. Lond. Math. Soc., series 3, 72 (1996) 261–280. MR 96k:11111
2. K. Soundararajan, Small gaps between prime numbers: The work of Goldston-Pintz-Yildirim
3. The Wikipedia entry on prime gaps