Bounded length intervals containing two primes and an almost-prime

Abstract

Goldston, Pintz and Yıldırım have shown that if the primes have ‘level of distribution’ θ for some θ > ½, then there exists a constant C(θ), such that there are infinitely many integers n for which the interval [n, n + C(θ)] contains two primes. We show under the same assumption that, for any integer k ⩾ 1, there exist constants D(θ, k) and r(θ, k), such that there are infinitely many integers n for which the interval [n, n + D(θ, k)] contains two primes and k almost-primes, with all of the almost-primes having at most r(θ, k) prime factors. If θ can be taken as large as 0.99, and provided that numbers with 2, 3 or 4 prime factors also have level of distribution 0.99, we show that there are infinitely many integers n such that the interval [n, n + 90] contains two primes and an almost-prime with at most four prime factors. This extends a result of Pintz [‘Are there arbitrarily long arithmetic progressions in the sequence of twin primes?’ An irregular mind, vol. 21, Bolyai Society Mathematical Studies (János Bolyai Math. Soc., Budapest, 2010) 525–559].

Locations

• Bulletin of the London Mathematical Society - View
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• Oxford University Research Archive (ORA) (University of Oxford) - View - PDF
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