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Let $n>m\ge 2$ be positive integers and $n=(m+1)\ell +r$, where $0\le r\le m.$ Let $C$ be a subset of $\{0,1,\cdots ,n\}$. We prove that if $$\left|C\right|>\left\{\begin{array}{cc}\lfloor n/2\rfloor +1\hfill & \text{if}m\text{is}\text{odd},\hfill \\ m\ell /2+\delta \hfill & \text{if}m\text{is}\text{even},\hfill \end{array}\right.$$ where $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$ and $\delta$ denotes the cardinality of even numbers in the interval $[0,min\{r,m-2\left\}\right]$, then $C-C$ contains a power of $m$. We also show that these lower bounds are best possible.