On the sum of two powered numbers

TL;DR

The paper proves that for any fixed >1/2, large natural numbers can be expressed as the sum of two numbers from a specific set defined by a divisor condition, nearly reaching the theoretical limit.

Contribution

It establishes a near-optimal threshold for representing large numbers as sums of two elements from a divisor-based set, extending understanding of additive number theory.

Findings

For >1/2, all sufficiently large numbers are sums of two set elements.

The result is nearly optimal with respect to the threshold .

The set  is defined by a divisor condition related to square-free divisors.

Abstract

Fix a positive real number $\theta$. The natural numbers $m$ with largest square-free divisor not exceeding $m^\theta$ form a set $\mathscr{A}$, say. It is shown that whenever $\theta>1/2$ then all large natural numbers $n$ are the sum of two elements of $\mathscr{A}$. This is nearly best possible.