Video narration
Could a final checklist of every prime number really exist? Assume the checklist is complete: p-one, p-two, and so on, up to p-n. Now multiply every item on the checklist and add one: N equals p-one times p-two, continuing through p-n, plus one. Take any prime from the list. It divides the product, so dividing N by it leaves remainder one. Therefore, none of the listed primes divides N. Here is the turn: N is greater than one, so it has a prime divisor. That divisor cannot be on the checklist, because no listed prime divides N. This is a proof by contradiction: assume the checklist is complete, then it proves itself incomplete. Could a final checklist exist? No. Any finite checklist creates a number with a prime divisor outside the checklist, so the list can never end. There must be infinitely many prime numbers.