Another general observation is that the product of each row is one. This follows from the observations that the rows are bookended by \$$\frac{1}{n}\$$ and \$$\frac{n}{1}\$$, and the fact that \$$f(\frac{p}{q}) = \frac{q}{p + q - 2(p \mod q)}\$$. I suppose that to complete the proof, you'd need to show that the RHS is still in lowest terms whenever the LHS is. But if \$$(p, q) = 1\$$, then so is \$$(q, p + q - 2r) \$$, where \$$r = p \mod q\$$, so we're ok.