Prime gaps

From Polymath1Wiki

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(logp_n).

A recent result of Goldston-Pintz-Yildirim shows that there exist infinitely many n for which the gap is as small as o(logp_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(log²p_n) in size. If so, this resolves the finding primes project positively. However, the best upper bound on the prime gap is assuming the Riemann hypothesis, and otherwise (a result of Baker, Harman, and Pintz; an earlier bound of was obtained by Baker and Harman.).

Rankin showed that the prime gap can be as large as .

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