While reading a very old book on diophantine equations, I came across this exercise:
Find an infinite number of positive integer solutions of the equations $$x^2 + y^2 = u^2$$ $$y^2 + z^2 = v^2$$ $$z^2 + x^2 = w^2$$
I have found a few solutions by hand, for example $x=240$, $y = 117$, $z = 44$, and trivially multiples will also produce solutions, but I assume that the book is really asking for solutions where there is no common factor of $x$, $y$, $z$.
I have spent a few hours trying to get something from the standard parametric solutions of $x^2 + y^2 = z^2$ without success, and wondered if anyone has any insight they can offer.
Clearly this is (slightly) connected with the problem of finding an integer cuboid with all the face diagonals integral, and the main diagonal also integral, which I assume is still an open problem.