Prime Plus a Non-Square-Free Integer: An Elementary Approach

Abstract

Lee and O'Clarey recently conjectured that every integer $n>24$ can be written as the sum of a prime and a positive integer that is not square-free. They proved this for odd $n$, and for all $n>24$ assuming the generalised Riemann hypothesis for Dirichlet $L$-functions. We prove their conjecture unconditionally for every integer $n>24$ that is not divisible by $997\#$, where $997\#=\prod_{p\leq 997}p$. In particular, any counterexample must be divisible by every prime up to $997$, and hence exceeds $10^{415}$. The proof uses an elementary congruence argument together with finite computation.