Infinitely Many Primes

Euclid's idea, recast as a contradiction, proves primes go on forever. Assume the opposite: there are only finitely many primes, say p₁, p₂, ..., pₙ. Multiply them all together and add 1: N = p₁ × p₂ × ... × pₙ + 1. This new number N is not divisible by any of the p's — the remainder is always 1.

(Euclid's own proof was actually direct: take any finite list of primes, build N the same way, and show it forces a brand-new prime. He never assumed the list was complete — that contradiction framing is a modern reworking.)

So N is either prime itself (a new prime not in our list) or divisible by some prime not in our list. Either way, our "complete" list of primes was incomplete. Contradiction. Therefore no finite list can contain all primes — they must be infinite.