The primes never run out
Suppose you claim to hold ALL the primes in a finite list. Multiply them together and add 1. That new number leaves remainder 1 when divided by every prime on your list — yet it must have SOME prime divisor. So your list was incomplete. Any finite list fails, which is exactly what 'infinite' means. It's one of the most elegant arguments in all of mathematics: it doesn't find primes, it shows your net can never catch them all.
A wanted-poster wall claiming to show every outlaw; the proof is a machine that studies the wall and always sketches a face that isn't on it.
List: 2, 3, 5. Then 2×3×5 + 1 = 31 — prime, and not on the list. List: 2,3,5,7,11,13 → product+1 = 30031 = 59 × 509: not prime itself, but its prime factors 59 and 509 are new. Either way you win.
Beyond its beauty, it guarantees cryptography never runs out of raw material — there are always fresh large primes to build keys from.
Level 1 The precise statement
There are infinitely many prime numbers.
Level 2 The proof
Level 3 What it stands on (4 direct)
- Infinite set (definition)
- Integers form a commutative ring (axiom)
- Divisibility (definition)
- Prime number (definition)
- Divisibility respects linear combinations (lemma)
- Well-Ordering Principle (axiom)
- Every integer > 1 has a prime divisor (lemma)
- Infinitude of primes (Euclid) (theorem)
Level 4 The verified record
This page is generated from a machine-checked node. The kernel confirms its dependencies resolve, nothing is circular, and it grounds in axioms (foundation: ZFC). The content hash below makes tampering evident.
References: Euclid, Elements, Book IX, Proposition 20
1 downstream result would collapse with it. See the blast radius on the graph →