Question 86
Let $S$ be the set of all positive integers $n$ such that $n^4$ has a divisor in the range $n^2+1, n^2+2, \dots, n^2+2n$. Prove that there are infinitely many elements of $S$ of each of the forms $7m$, $7m+1$, $7m+2$, $7m+5$, $7m+6$ and no elements of $S$ of the form $7m+3$ or $7m+4$, where $m$ is an integer.