Small differences between prime numbers.
About: This article is published in Michigan Mathematical Journal. The article was published on 01 Jan 1988. and is currently open access. The article focuses on the topics: Prime triplet & Prime k-tuple.
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Citations
Bounded gaps between primes
Abstract: It is proved that
lim inf n?8 (p n+1 -p n )<7×10 7 , where p n is the n -th prime.
Our method is a refinement of the recent work of Goldston, Pintz and Yildirim on the small gaps between consecutive primes. A major ingredient of the proof is a stronger version of the Bombieri-Vinogradov theorem that is applicable when the moduli are free from large prime divisors only, but it is adequate for our purpose
Primes in tuples I
TL;DR: In this article, it was shown that there are infinitely often primes differing by 16 or less in the Elliott-Halberstam conjecture and that there exist consecutive primes which are closer than any arbitrarily small multiple of the average spacing.
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Primes in tuples II
TL;DR: In this paper, it was shown that there are infinitely often primes differing by 16 or less in the Elliott-Halberstam conjecture and that there exist consecutive primes which are closer than any arbitrarily small multiple of the average spacing.
Small gaps between prime numbers: The work of Goldston-Pintz-Yildirim
TL;DR: Goldston, Pintz, and Yildirim as discussed by the authors showed that there are infinitely many prime numbers for which the gap to the next prime is as small as we want compared to the average gap between consecutive primes.
Small gaps between primes or almost primes
TL;DR: In this paper, it was shown that for numbers with two distinct prime factors, the bound can be improved to 6 by a generalization of the Elliott-Halberstam Conjecture.
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