Well I was just solving a problem that has an alternative (non-elementary) solution using the elliptic curves. Seeking for an elementary solution, I had to find all possible values of $xy$ such that there exist $x,y,z\in\mathbb{Z}$ so that :
$$z|xy+1\quad,\quad y|16x^2z^2+1$$
$$x,y,z\quad\text{are pairwise coprime}$$
Well not the first time I encountered with a so-called "interwined" system of modular equiations (The term is just made up by myself since in these systems there exists a variable that is in the LHS of one equation and in RHS of another equation simultaneously) but as far as I recall, I couldn't solve any of them in the general case. (Under some special conditions one may be able to do some p-adic , inequalities or things like that but it's not useful here) All I could try to do for solving this system was to try to isolate $x$ in the two equations and form a CRT-suitable system of modular equiations but then I'm fairly sure that the CRT cannot be solved in the general case and this approach will fail. I haven't had any other ideas yet.
So I'd want to know if there is a general approach to these systems of equations.
Any elementary/non-elementary solutions/approaches are useful.
P.S. In case you're wondering about the original problem it asks to find all integers $n$ such that $n^3+n+1$ is a perfect square.
Edit : All possible answers to the problem seem to be $$(x,y,z)=(0,\pm1,\pm1),(1,1,0),(-1,-1,0),(\pm1,\pm305,\pm18),(\mp18,\pm17,\pm1)$$ which are equivalent to the answers $$(a,b)=(0,\pm1),(\pm72,\pm611)$$ for the diophantine equation $a^3+a+1=b^2$ (the original statement of the problem) due to the solution with the elliptic curves.
