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 $$m = (a-1)k-1,$$ $$n = \frac{(a-1)k}{a} - 1.$$