SMALL GAPS BETWEEN PRIMES OR ALMOST PRIMES

By D. Goldston, S. Graham, Janos Pintz, C. Yıldırım. 2005

55 citations recorded by the source corpus.

Let p n denote the n th prime. Goldston, Pintz, and Yildirim recently proved that li m in f (pn+1 ― p n ) n→∞ log p n = 0. We give an alternative proof of this result. We also prove some corresponding results for numbers with two prime factors. Let q n denote the n th number that is a product of exactly two distinct primes. We prove that lim inf(qn+1―q n ) n→∞ 26. If an appropriate generalization of the Elliott-Halberstam Conjecture is true, then the above bound can be improved to 6.

Subfields: Algebra and Number Theory, Geometry and Topology, Theoretical Computer Science

View Paper: https://www.semanticscholar.org/paper/d6cdf2d7f91799ac22dbbb797f97382ca56d4cea

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In Algebra and Number Theory

In Geometry and Topology

In Theoretical Computer Science

Citations in the corpus

Bounded gaps between primes (2014, methodology): “There are two methods to calculate the main terms, one more analytic ( gpy [15]), the other, ( sound [34], ggpy [16]), more combinatorial.”
Variants of the Selberg sieve, and bounded intervals containing many primes (2014, methodology): “s k‚ 1 i1 Spfi;jq Spfi;j1q€# which makes Theorem3.5(i) available for use. Indeed, for any jPt1;:::;Juand i1;:::;k, we have from (87) that Spfi;jq⁄p1 "q # 2 €#€1 Polymath Page 39 of79 and so by (20) we have pnqpn hkq¥ p ‚ jPJ 1 2 ‚ jPJ 2 qcjf 1;j pn h1q:::f k 1;j pn hk 1qfk;jp0q ‚ j1PJ 1 cj1f 1;”
Primes in tuples I (2005, methodology): “A more algebraic discussion of d may also be found in [14], [15].”
Primes in intervals of bounded length (2014, unknown): “There are several techniques in analytic number theory that allow one to get accurate estimates for such sums, two more analytic ([23], [58]), the other more combinatorial ([62], [24]).”