@MartinBüttner: Indeed. Confirmed that the former vector has magnitude `0.6726813227673505132725932e8` and the latter has magnitude `0.6726813227673505876019945e8`, differing in the 17th digit (would be the 16th in a normalized float).
I'm not a great student of algebraic geometry, but perhaps there's an efficient way to determine all integer solutions to the Diophantine equation `x^2+y^2-w^2-z^2=n`, in which case you could simply step `n` over a reasonable range of values to determine different potential sets of false positives.