Euclid proved more than two thousand years ago that the list of prime numbers never comes to an end.

The longer version

Euclid's argument builds a witness. Multiply every prime you have found so far, then add one. The result leaves a remainder of one when divided by any prime on your list, so either it is prime itself or it has a prime factor you had not recorded. Either way the list was incomplete.

The reasoning appears in the Elements, compiled around three hundred BCE, and it still reads as a model of economy: no computation, no case checking, just one construction that works for any finite collection. Modern searches for record primes use very different machinery, yet they only ever extend a supply whose endlessness was settled long before.

Where this entry sits

This section collects odd corners of mathematics, from stubborn prime numbers to counting systems older than modern schools. Expect famous constants, probability puzzles, shapes that break everyday intuition, and numbers far too large to picture. The full list sits on the Numbers and Math page, where every entry is listed in order.