Schor’s algorithm does work on a classical computer.  Shor’s algorithm contains many steps, all of which run quickly on a classical computer except for the period finding step, which runs slowly and can only be efficiently evaluated by a quantum computer. There are various ways to factor classically, but they all require exponential time. The best current algorithm is the general number field sieve. Quantum computers, though, can factor in polynomial time and fully utilize the benefits from Shor’s algorithm.
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