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

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 $n > 24$.