On the representation of integer as sum of a square-free number and a prime of special type

TL;DR

This paper proves the existence of infinitely many integers that can be expressed as the sum of a square-free number and a specially constrained prime, extending understanding of additive number theory involving primes and square-free numbers.

Contribution

It establishes the infinitude of integers representable as a sum of a square-free number and a prime satisfying a specific Diophantine approximation condition, a novel result in additive number theory.

Findings

Infinitely many integers can be written as the sum of a square-free number and a prime with a certain approximation property.

The prime involved satisfies a Diophantine inequality involving an irrational multiple.

The result advances understanding of the additive structure involving primes and square-free numbers.

Abstract

We prove that there are infinitely many integers, which can represent as sum of a square-free integer and a prime $p$ with $||\alpha p+\beta||<p^{-1/10}$, where $\alpha$ is irrational.