Triples of almost primes

Triples of almost primes

The k -tuple conjecture of Hardy and Littlewood predicts that there are infinitely many primes p such that p + 2 and p + 6 are primes simultaneously. In this paper, we prove that there are infinitely many primes p such that Ω( p + 2) ⩽ 3 and Ω( p + 6) ⩽ 6, where Ω( n ) denotes the total number of pr...

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Veröffentlicht in:Science China. Mathematics 2023-12, Vol.66 (12), p.2779-2794
Hauptverfasser: Li, Jiamin, Liu, Jianya
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Applications of Mathematics
Mathematics
Mathematics and Statistics
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description The k -tuple conjecture of Hardy and Littlewood predicts that there are infinitely many primes p such that p + 2 and p + 6 are primes simultaneously. In this paper, we prove that there are infinitely many primes p such that Ω( p + 2) ⩽ 3 and Ω( p + 6) ⩽ 6, where Ω( n ) denotes the total number of prime divisors of an integer n . We also prove a better conditional result, with the above Ω( p + 6) ⩽ 6 replaced by Ω( p + 6) ⩽ 3, under the Elliott-Halberstam conjecture.
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Mathematics and Statistics
title Triples of almost primes
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