Question 131
Let $S=\{1,4,8,9,16, \ldots\}$ be the set of perfect powers of integers, i.e. numbers of the form $n^{k}$ where $n, k$ are positive integers and $k \geq 2$. Write $S=\left\{a_{1}, a_{2}, a_{3} \ldots\right\}$ with terms in increasing order, so that $a_{1}\lt a_{2}\lt a_{3} \cdots$. Prove that there exist infinitely many integers $m$ such that $9999$ divides the difference $a_{m+1}-a_{m}$.