Why 1 is not a prime
This is a convention with a reason. If 1 were prime, then 6 = 2×3 = 1×2×3 = 1×1×2×3 = … and uniqueness of factorization — the crown jewel — would be false as stated. Mathematicians didn't discover 1 isn't prime; they CHOSE the definition that makes the deepest theorem clean. Counterexamples like this document the sharp edges of definitions.
A dictionary deliberately excluding the blank page from the list of words: including it would break every sentence count.
Pretend 1 is prime. Then 6 has factorizations 2×3 and 1×2×3 — different lists, so 'unique factorization' dies. Excluding 1 saves the theorem.
It teaches the meta-lesson: definitions are engineered, and a good counterexample is the documentation of WHY they are engineered that way.
Level 1 The precise statement
1 is not prime: the definition requires p > 1, so 1 is excluded. This is what makes prime factorization unique.
Level 3 What it stands on (1 direct)
- Integers form a commutative ring (axiom)
- Divisibility (definition)
- Prime number (definition)
- 1 is not prime (counterexample)
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.
Nothing yet — this is a frontier result.
Nothing depends on it yet, so its failure would be contained.