On the Sum of a Prime and a Number that is not Square-Free

TL;DR

The paper proves that large integers can be expressed as the sum of a prime and a non-square-free number, with unconditional results for odd integers and conditional results under GRH.

Contribution

It establishes the sum representation for large integers with new unconditional and conditional proofs, advancing understanding of additive number theory.

Findings

Unconditionally holds for odd n > 24.

Under GRH, holds for all n > 24.

Discusses obstructions to unconditional proof for all n > 24.

Abstract

We prove that every sufficiently large integer $n$ can be written as the sum of a prime and an integer that is not square-free. In addition, we expect this result holds for every $n > 24$ and prove two results to support this claim. First, we prove the result holds unconditionally for every odd $n > 24$. Second, assuming the Generalised Riemann Hypothesis for Dirichlet $L$-functions, we prove the result holds for every $n > 24$. We also discuss the obstruction which prohibits us from proving the result unconditionally for every $n > 24$.