There is no final prime.

A prime is a whole number greater than one whose only positive divisors are one and itself. Start a list: 2, 3, 5, 7, 11. Perhaps there is an enormous last prime somewhere. To test that possibility, imagine that someone hands you a complete finite list of every prime. Do not try to search beyond it. Use the list against its own claim.

Multiply all the listed primes together and add one. Call the result N. Dividing N by any prime on the list leaves a remainder of one, because the product itself was exactly divisible by each listed prime. None of them divides N. Yet N is greater than one, so it must have a prime divisor. That divisor is missing from the supposedly complete list. The assumption of completeness has contradicted itself.

The new number does not have to be prime. For example, multiplying 2, 3, 5, 7, 11 and 13, then adding one, gives 30,031, which is 59 × 509. The argument works because a new prime factor is enough. This is why a proof differs from a promising pattern: it does not depend on the next example being lucky. Associated with Euclid, the reasoning reaches an unlimited conclusion without writing an unlimited list. A small operation—add one—makes the edge of any finite list fail. Sometimes the way to learn what cannot end is to imagine that it already has.