REPRESENTING AN INTEGER AS THE SUM OF A PRIME AND THE PRODUCT OF TWO SMALL FACTORS. Issue 2 (6th March 2021) - Baker, Roger; Harman, Glyn
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.