When following the procedure on wikipedia for wheel factorization, I seem to have stumbled into a problem where the prime number 331 is treated as a composite number if I try to build a 2-3-5-7 wheel.
With 2-3-5-7 wheel, 2*3*5*7=210. So I setup a circle with 210 slots and go through steps 1-7 without any issues. Then I get to step 8 and strike off the spokes of all multiples of prime numbers, I eventually strike off the spoke rooted at 121, which is a multiple of 11, which is a prime. For the spoke rooted at 121, 121 + 210 = 331. Unfortunately, 331 is a prime number.
Is the procedure on Wikipedia incorrect?
Or did I misunderstand the procedure, and should have only struck out spokes that are multiples of 2, 3, 5, and 7, but not any of the other primes less than 210?
Wikipedia is correct.
331 is in the 1 spoke of the wheel. The spoke is not shaded, so 331 is potentially prime. And in fact, it is prime.
121 is also in the 1 spoke of the wheel, so 121 is potentially prime. That is, it is not eliminated as a prime by the wheel. However, it is not prime.
The wheel doesn't allow you to make any inference about the primality of 331 based on the non-primality of 121. Sorry.
I have an implementation of wheel factorization at my blog, if you want to look at it.
Yes, you are only allowed to strike off the spokes that are multiples of 2, 3, 5 and 7. In fact, 121 which is a multiple of 11, is relatively prime to 210. So the numbers on the 121 spoke can be either prime or composite.
来源：https://stackoverflow.com/questions/8341295/2-3-5-7-wheel-factorization-seems-to-skip-prime-number-331