Conjecture to establish if a number is not prime

If the quotient of a/b has a period of "c" figures and (b-1)/c results to be a non-integer number, then "b" is NOT a prime number. "a" and "b" are two integers, primes between them.

For instance, from 2/57 results the quotient 0,035087719298245614 which has a period of 18 figures and because the result of (57-1)/18 is 3,1 which is a non-integer, then 57 is not prime.

I have to mention that this conjecture doesn't determine if “b” is prime but if “b” is not prime, since it does not succeed in finding all numbers that are not prime.

The non-prime numbers that the conjecture does not find on an interval between 3 and 5000 (from which all multiples of 2 and 5 have been excluded) are: 9 - 33 - 91 - 99 - 259 - 451 - 481 - 561 - 657 - 703 - 909 - 1233 - 1729 - 2409 - 2821 - 2981 - 3333 - 3367 - 4141 - 4187 - 4521 - …

The Carmichael numbers are a subset of the whole non-prime numbers which the conjecture can not identify.

I would appreciate any suggestions or demonstrations regarding this conjecture. Thank you all in advance.