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For any finite list of primes, there exists at least one prime that is not on that list.
Letting P be the product of all primes on the list, if P + 1 is not prime, then it has a prime factor not on the list.
A prime factor of P + 1 cannot divide P, but all primes on the list can divide P.
In order to divide both P and P + 1, the prime factor would also need to divide 1, and no prime number divides 1.
Letting P be the product of all primes on the list, if P + 1 is prime, then it is a prime number not on the list.