Euclid proved the primes never run out — 2,300 years ago

Primes — numbers like 2, 3, 5, 7, 11 divisible only by 1 and themselves — grow rarer as you count higher. Yet they never stop. Around 300 BC, in Book IX of his Elements, Euclid proved there are infinitely many of them, in one of the first arguments ever to use proof by contradiction.

Suppose the primes were finite, a complete list. Multiply them all together and add 1. The resulting number leaves a remainder of 1 when divided by every prime on the list — so none of them divides it.

That means a prime is missing from your supposedly complete list, a contradiction. The list can never be finished.

Euclid couldn’t write ‘infinitely many’ — the Greeks phrased it as primes being more than any assigned multitude.