@nprofile1q... A followup on this one. Turns out that Shor needs so many qubits because it needs to compute x^r mod n for all r in range 1..n-1. Shor does this with quantum magic and lots of qubits.
The giant reduction in qubits for the claimed breakthrough is that they compute these classically and then load them up into the quantum computer. Of course that saves on qubits, but only at the expense of exponential running time. So, not actually a breakthrough, just more trickery.