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

TL;DR

This paper proves that large integers can be expressed as the sum of a prime and a product of two small factors, improving previous bounds and introducing a new sieve technique.

Contribution

It advances number theory by lowering the exponent c for representing integers as prime plus small factors and introduces the 'intersecting two sieves' method.

Findings

Every large n can be written as p + ab with p prime and a,b small

Improves previous bound c > 3/4 to c > 0.55

Introduces a new sieve technique called 'intersecting two sieves'

Abstract

Let c > 0.55. Every large n can be written in the form p +ab, where p is prime, 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. We introduce the idea of 'intersecting two sieves' as a tool in the proof of this result.