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Let $[\, \cdot\,]$ be the floor function. In this paper, we show that when$1<c<\frac{82}{79}$, then every sufficiently large positive integer $N$ can berepresented in the form \begin{equation*} N=[p^c]+[m^c]\,, \end{equation*}where $p$ is a prime and $m$ is a square-free.

A binary additive equation with prime and square-free number