Every number contains an atom
Take any number > 1 and keep splitting it into smaller factors; the process must stop (numbers can't shrink forever — that's well-ordering), and where it stops is a prime. So no number escapes the primes: each one is either prime itself or divisible by one.
Keep cutting a molecule into smaller pieces and you must eventually hit an atom.
Take 60 → 6 × 10 → the smallest divisor > 1 of 60 is 2, and 2 is prime. Take 49 → smallest divisor > 1 is 7 — prime. It never fails.
This is the entry point of both landmark theorems here: Euclid's proof needs it to find a NEW prime, and the Fundamental Theorem uses it to start every factorization.
Level 1 The precise statement
Every integer n > 1 has at least one prime divisor.
Level 2 The proof
Level 3 What it stands on (3 direct)
- Well-Ordering Principle (axiom)
- Integers form a commutative ring (axiom)
- Divisibility (definition)
- Prime number (definition)
- Every integer > 1 has a prime divisor (lemma)
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: peano). The content hash below makes tampering evident.
3 downstream results would collapse with it. See the blast radius on the graph →