REPRESENTING AN INTEGER AS THE SUM OF A PRIME AND THE PRODUCT OF TWO SMALL FACTORS. Issue 2 (6th March 2021)

• Record Type: Journal Article
• Authors: Baker, Roger; Harman, Glyn
• Is Part Of: Mathematika. Volume 67:Issue 2(2021)
• Page Start: 447
• Page End: 467
• Publication Date: 2021-03-06
• Subjects: 11N13 (primary); Mathematics -- Periodicals; 510.5
• DOI: 10.1112/mtk.12084
• Languages: English
• ISSNs: 0025-5793

Abstract

Abstract: Let ε > 0 . We show that every large integer n may be written in the form n = p + a b, where a, b ⩽ n 1 2 − δ for a positive absolute constant δ, and a b ⩽ n 0.55 + ε . This sharpens a result of Heath‐Brown ( Math. Proc. Cambridge Philos. Soc . 89 (1981), 29–33). The improvement depends on a lower bound version of Bombieri's theorem in short intervals. In establishing such a result, we shall need to "intersect" two lower bound prime‐detecting sieves, and we give a more general discussion on this point which may have further applications.