How to solve this Diophantine equation？ [closed]

$$ \frac{1}{m^{2}} - \frac{1}{n^{2}} = \frac{1}{p^{2}} - \frac{1}{q^{2}} \neq 0 $$

I guess $\left | m \right | = \left | p \right | $ and $\left | n \right | = \left | q \right | $.

Is there any way to exclude other situations?

Solution 1:

No, there are counterexamples, e.g., take $$ (m,n,p,q) = (100,156,65,75). $$ Note that the equation can be rewritten as $$ \frac{1}{m^{2}} + \frac{1}{q^{2}} = \frac{1}{p^{2}} + \frac{1}{n^{2}} $$ This has infinitely many solutions with $\gcd(m,q,p,n)=1$, e.g., among $$ (m,q,p,n)=(2k^2(2k^2-1),2k^2-1,k(2k^2-1),2k^2), k\in \mathbb{Z}_{+}. $$