Additive problems with almost prime squares

TL;DR

This paper proves that large integers and shifted primes can be expressed as sums involving primes, smooth numbers, and almost prime squares, using advanced analytic and algebraic techniques to achieve uniform results.

Contribution

It introduces new methods combining analytic, automorphic, and algebraic approaches to study representations involving almost prime squares and restricted quadratic forms.

Findings

Every large integer is a sum of a prime and two almost prime squares.

Shifted primes p-1 can be expressed as sums of two almost prime squares.

The number of such representations matches the expected order of magnitude.

Abstract

We show that every sufficiently large integer is a sum of a prime and two almost prime squares, and also a sum of a smooth number and two almost prime squares. The number of such representations is of the expected order of magnitude. We likewise treat representations of shifted primes p-1 as sums of two almost prime squares. The methods involve a combination of analytic, automorphic and algebraic arguments to handle representations by restricted binary quadratic forms with a high degree of uniformity.