There is a large theory of quadratic residues. As far as I know there is no comparably large theory of cubic residues or history of studying them. A search on this site finds hundreds of questions about quadratic residues (which even has its own tag, quadratic-residues), and only 31 about cubic residues.
What mathematical features of quadratic residues lead to their having an important and rich theory, whereas there seems to be relatively little to say about cubic residues?
I have a couple of thoughts:
- The Brahmagupta-Fibonacci identity ensures that quadratic forms have interesting multiplicative properties; there is no corresponding identity for cubes.
- The Euclidean metric involves second powers and not third powers, so questions about geometric properties of $\Bbb Z^n$ lattices are more likely to depend on properties of second powers than third powers.
But perhaps these are two faces of the same underlying issue. I think they tie together in the fact that while $|z|=\sqrt[3]{x^3+y^3}$ can be used as a norm on $\Bbb C$, it lacks the important property that $|z_1z_2| = |z_1||z_2|$.
Still I feel like I'm missing the bigger picture. What's the bigger picture?