Consider the equation $$\frac{a^2}{a^2-1} \cdot \frac{b^2}{b^2-1} = \frac{c^2}{c^2-1}.$$ Of course, there are solutions to this like $(a,b,c) = (9,8,6)$. Is there any known approximation for the number of solutions $(a,b,c)$, when $2 \leq a,b,c \leq k$ for some $k \geq 2.$

More generally, consider the equation $$\frac{a_1^2}{a_1^2-1} \cdot \frac{a_2^2}{a_2^2-1} \cdot \ldots \cdot \frac{a_n^2}{a_n^2-1} = \frac{b_1^2}{b_1^2-1} \cdot \frac{b_2^2}{b_2^2-1}\cdot \ldots \cdot \frac{b_m^2}{b_m^2-1}$$ for some natural numbers $n,m \geq 1$. Similarly to the above question, I ask myself if there is any known approximation to the number of solutions $(a_1,\ldots,a_n,b_1,\ldots,b_m)$, with natural numbers $2 \leq a_1, \ldots, a_n, b_1, \ldots, b_m \leq k$ for some $k \geq 2$. Of course, for $n = m$, all $2n$-tuples are solutions, where $(a_1,\ldots,a_n)$ is just a permutation of $(b_1,\ldots,b_n)$.

likemean in "solutions ... like $(a,b,c)=(9,8,6)$"? $\endgroup$ – Gerry Myerson May 6 at 12:29`(a,b,c)`

=`(9,8,6),(26,15,13),(55,24,22),(99,35,33),(50,49,35),(161,48,46),(120,55,50),(244,63,61),(351,80,78),(485,99,97),...`

$\endgroup$ – Dmitry Ezhov May 6 at 14:28`2..1000`

give all $n$ in`3..31`

$\endgroup$ – Dmitry Ezhov May 7 at 5:03