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.
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