Every number has exactly one recipe
Two separate miracles in one theorem. Existence: you can always factor down to primes (keep splitting; well-ordering says you stop). Uniqueness: no matter HOW you split, you end with the same primes — guaranteed by Euclid's lemma, which stops a prime from hiding in different factors on different runs. Together they make primes a coordinate system: every number is uniquely an address written in primes.
Every substance has one chemical formula. Water is H₂O whether you condensed it or melted it; 60 is 2²·3·5 whether you split it as 6×10 or 4×15.
60 = 6 × 10 = (2×3) × (2×5) = 2²·3·5. Also 60 = 4 × 15 = (2×2) × (3×5) = 2²·3·5. Different routes, same primes — that's the theorem working.
Unique factorization is the license behind gcd/lcm reasoning, modular arithmetic, and RSA: 'the' prime factorization exists to be secret only because it is unique.
Level 1 The precise statement
Every integer n > 1 can be written as a product of primes, and this factorization is unique up to the order of factors.
Level 2 The proof
Level 3 What it stands on (4 direct)
- Axiom of Induction (axiom)
- Integers form a commutative ring (axiom)
- Divisibility (definition)
- Prime number (definition)
- Well-Ordering Principle (axiom)
- Greatest common divisor (definition)
- Bezout's identity (lemma)
- Euclid's lemma (lemma)
- Every integer > 1 has a prime divisor (lemma)
- Fundamental Theorem of Arithmetic (theorem)
Level 4 The verified record
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References: Hardy & Wright, An Introduction to the Theory of Numbers
Nothing yet — this is a frontier result.
Nothing depends on it yet, so its failure would be contained.