[lutusp]

> I think a given divisor does not need to be prime; but it must not be divisible by an initial prime.

But that's how prime is defined -- indivisible by any other numbers except 1. If you statement is true -- that a given number "must not be divisible by an initial prime", that means the number is itself prime.

Positive integers fall into precisely two categories:

1. Not divisible by any smaller numbers except 1.

2. Divisible by one or more smaller numbers.

Those in category (1) are prime. Those in category (2) are composite. There is no third possibility.

[hyp0]

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".)