Prime numbers thin out but never run out

Euclid proved over two thousand years ago that there are infinitely many primes. They get rarer among large numbers, but the supply is truly endless. Primes are the numbers divisible only by themselves and one, and they become noticeably rarer as numbers get larger, since there are simply more possible smaller factors around to rule a number out. Despite that thinning, the supply never actually ends. Euclid proved this over two thousand years ago with an elegantly simple argument: assume there were only finitely many primes, multiply them all together and add one, and the result must either be prime itself or divisible by some prime missing from the original list, a contradiction either way. That guarantees infinitely many primes exist. Modern research has moved on to subtler questions this proof doesn't touch, like exactly how primes are spaced, including the still-unproven twin prime conjecture about pairs of primes just two apart.

Prime numbers thin out but never run out

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