I'd like to find integral solutions to the equation
$2x^2 -3xy + y^2 \equiv 0 \mod n $
where $n$ is a given composite, for example, $n = 16807708473783470801$ (I prefer solutions that work for any $n$). I don't know the factorization of $n$, though I do know it is a product of two large, distinct primes.
Any thoughts? I know being able to quickly find solutions to $x^2 \equiv y^2 \mod n$ is equivalent to factoring, so I'm not super hopeful for provably fast algorithms for my problem - I'd be happy for heuristics, links to Mathematica documentation, etc.
Would the problem become any easier or richer if I replaced the quadratic with a cubic? In that case I'd expect techniques from the theory ellipic curves to play a role..