JohnDoe wrote:Let n be a positive integer. Let 2, 3, ..., pk be a listing of all primes less than or equal to n.
Then, for i in the range [1,n], the positive integer
Q(i) = 2*3*...*pk + i
Did you mean for i ∈ [2,n}?
Also, I don't see how this can be extended to provide a counterexample for III. The density of these composite stretches doesn't seem high enough to allow for choosing n+1 composite numbers in [1, 2n]. Obviously, there do exist n+1 composite numbers in [1, 2n], but that's more a consequence of the prime number theorem than of this particular fact.
You don't even actually need the full power of the PNT. In [1, 2n], n-1 integers are composite multiples of 2. All you have to do is find 2 more non-prime numbers in [1, 2n] and you're set.