You can't have prime factorization without having a number to factor.
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The proof that there is no largest prime:Assume that there are a finite number of primes for the sake of contradiction. Then, there should be a number P that equals p1p2p3...pn+1. P is either prime or not prime (composite). If it is prime, we just show that P is larger than the largest prime in the list. If it's not prime, it must be composite. Composite always has at least one factor that is prime, but since P is not divisible by any prime in the list, the unknown prime factor(s) must be something not in the list, this also shows that there is a prime larger than the largest prime in the list. Both cases show that no matter how large a list of prime numbers, there will be always at least one larger prime outside of that list.
1, 31 (31 is a prime!)
23 29.
79 83 89 97.
They are 23 and 29.