  We construct simplified quantum circuits for Shor's order-finding algorithm
for composites N given by products of the Fermat primes 3, 5, 17, 257, and
65537. Such composites, including the previously studied case of 15, as well as
51, 85, 771, 1285, 4369,... have the simplifying property that the order of a
modulo N for every base a coprime to N is a power of 2, significantly reducing
the usual phase estimation precision requirement. Prime factorization of 51 and
85 can be demonstrated with only 8 qubits and a modular exponentiation circuit
consisting of no more than four CNOT gates.
