Remarks on additive representations of natural numbers

TL;DR

This paper investigates additive representations of natural numbers using square-free integers and primes, providing new bounds and generalizations that improve upon previous results in number theory.

Contribution

It introduces new lower bounds for representations of numbers as sums involving square-free integers and primes, extending previous work with broader cases and tighter estimates.

Findings

Established a new lower bound for the number of such representations.

Generalized previous results to more cases including small primes and short intervals.

Provided a Goldbach-type upper bound result.

Abstract

For two relatively prime square-free positive integers $a$ and $b$, we study integers of the form $a p+b P_{2}$ and give a new lower bound for the number of such representations, where $a p$ and $b P_{2}$ are both square-free, $p$ denote a prime, and $P_{2}$ has at most two prime factors. We also consider some special cases where $p$ is small, $p$ and $P_2$ are within short intervals, $p$ and $P_2$ are within arithmetical progressions and a Goldbach-type upper bound result. Our new results generalize and improve previous results.