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Let c > 0.55. Every large n can be written in the form p +ab, where p isprime, a and b are significantly smaller than x^1/2 and ab is less than n^c.This strengthens a result of Heath-Brown, which has the requirement c>3/4. Weintroduce the idea of 'intersecting two sieves' as a tool in the proof of thisresult.

Representing an integer as the sum of a prime and the product of two small factors