Problem 4. Describe all positive integer solutions $(m, n)$ of the equation $8 m-7=n^{2}$ and give the first value of $m$ (if it exists) greater than 1959.
Solution. Let $m=k+1$, we have $8 k+1=n^{2}$, or equivalently, $8 k=(n-1)(n+1)$. Evidently, $n$ is odd and if we consider the integer $s=\frac{n-1}{2}$, we deduce that $k=\frac{1+s}{2} s$ is a triangular number. In fact, if $k$ is triangular (i.e., $m$ is the number following a triangular number) there is always a solution, and thus the solutions are $(m, n)=$ $\left(1+\frac{1+s}{2} s, \sqrt{1+4 s(s+1)}\right)$ using that $s$ is a non-negative integer. Similarly, it is easy to verify that the first triangular number greater than 1959 is 2016.