This is a follow-up to a question posted recently. Let $s_n = \sum_{r=1}^{n} \frac{1}{r(r+1)},$ where we take $s_0 = 0$.
The problem I am interested in is this: For fixed $k \geq 2$, find all solutions $(m,n)$ in nonnegative integers to the Diophantine equation $s_m - s_n = \frac{1}{k}.$
Current state of knowledge (see the original question):
$s_m - s_n$ can be expressed as $s_m - s_n = \frac{m-n}{(m+1)(n+1)},$
At least some of the solutions are given by taking each divisor $a$ of $k$ such that $a > 1$ and setting $m = (a-1)k-1,$ $n = \frac{(a-1)k}{a} - 1.$