Quantum trial factoring?
 

Probably a very dumb question, but can Grover's algorithm be used for trial factoring?

I don't see why not. By [url=https://en.wikipedia.org/wiki/Holevo%27s_theorem]Holevo's theorem[/url] you'd need at least n qubits to factor an n-bit number, and a factor p could be found in time [TEX]O(\sqrt{p})=O(\sqrt[4]{N}).[/TEX] (Just search for factors of length 20 bits, 40 bits, 80 bits, ..., n/4 bits, n/2 bits.) It's not as fast as QS or NFS though.