Considering the problem of finding lattice points $(x_1, x_2 ... x_n)$ that satisfy a quadratic law:
$F(x_1, x_2... x_n) = 0$
such that $F(x_1, x_2... x_n)$ is a second degree polynomial
It is known that for some simple examples such as:
$x_1^2 + x_2^2 = x_3^2$ (pythagorean rule)
$x_1x_2 = N$ (factorization of integers)
the problem can be solved in polynomial time on a classical or quantum machine. Is it possible that in general all single equation quadratic diophantine equations can be solved in polynomial time on a Quantum Machine? Is there any strong counterexample or example that sets serious doubt?