Bounded length intervals containing two primes and an almost‐prime. (6th March 2013) - Maynard, James
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].