What does a prime appearing more than once on the list have to do with anything? The proof does NOT say anything about a "factoring series" (unless you mean the purported "list of all primes" itself) or primes appearing once in a "factoring series". It simply use the fact that either a number

**is** a prime or it is divisible by a prime (which is the

**definition** of "prime" and "composite" numbers). If there were a finite number of primes,

, then none of those numbers divides

. Either that number is itself prime or it is divisible by a prime that is NOT on that list. In either case that list does not include all primes.

If a number has

as a factor, then it has P1 as a factor.