I think you're interpreting "initial primes" as all the primes up to p+1; but here, I defined it to be just the particular primes we happened to start with. There can be gaps.
I think a given divisor does not need to be prime; but it must not be divisible by an initial prime.
I will disprove by counter-example that not being divisible by an initial prime implies it is prime:
initial primes: 5 and 7
p = 5 * 7 = 35
p+1 = 36
36 has a few divisors. Lets take 18 as a "given divisor". 18 is not divisible by any of the "initial primes", 5 and 7. Yet 18 itself is not prime. (While it is divisible by the primes 2 and 3, but they aren't "initial primes".)
Comments
I think you're interpreting "initial primes" as all the primes up to p+1; but here, I defined it to be just the particular primes we happened to start with. There can be gaps.
I will disprove by counter-example that not being divisible by an initial prime implies it is prime:
36 has a few divisors. Lets take 18 as a "given divisor". 18 is not divisible by any of the "initial primes", 5 and 7. Yet 18 itself is not prime. (While it is divisible by the primes 2 and 3, but they aren't "initial primes".)