(2*3*5*7*11*13)-1 = 30029 is prime (2*3*5*7*11*13)+1 = 30031 = 509*59 (composite)
If q is the product of a set of primes P, then q+1 and q-1 are relatively prime to each p_i, i.e. (q+1) % p_i = 1, and (q-1) % p_i = p_i - 1. However there are no guarantees about the relative primality of q with respect to some prime p not in P.
If the p_i in P are consecutive starting with 2, then we do know that Q+-1 must either by prime itself, and thus larger than any prime in P, or a composite of 2 or more primes, each of which is larger than any prime in P.
Given the set of all known primes, we can always generate a new unknown prime, thus expanding our set, and therefore guaranteeing the set of all primes is infinite.
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Here is a counterexample for your 'proof':
(2*3*5*7*11*13)-1 = 30029 is prime
(2*3*5*7*11*13)+1 = 30031 = 509*59 (composite)
If q is the product of a set of primes P, then q+1 and q-1 are relatively prime to each p_i, i.e.
(q+1) % p_i = 1, and (q-1) % p_i = p_i - 1. However there are no guarantees about the relative primality of q with respect to some prime p not in P.
If the p_i in P are consecutive starting with 2, then we do know that Q+-1 must either by prime itself, and thus larger than any prime in P, or a composite of 2 or more primes, each of which is larger than any prime in P.
Given the set of all known primes, we can always generate a new unknown prime, thus expanding our set, and therefore guaranteeing the set of all primes is infinite.