The unbreakable numbers
Primes are the numbers you cannot split into a product of smaller pieces: 7 refuses to be a rectangle of dots other than a 1×7 line. Every other number is built by multiplying primes, which makes them the atoms of arithmetic.
Chemical elements. Water is H₂O; 12 is 2×2×3. Primes are the periodic table of numbers.
Is 29 prime? Try dividing by primes up to √29 ≈ 5.4: not divisible by 2, 3, or 5. Prime. Is 91? 91 = 7 × 13. Not prime.
Online security rests on primes: multiplying two huge primes is easy, but recovering them from the product is so hard that it protects credit cards and private messages worldwide.
Level 1 The precise statement
An integer p > 1 is prime iff its only positive divisors are 1 and p.
Level 3 What it stands on (1 direct)
- Integers form a commutative ring (axiom)
- Divisibility (definition)
- Prime number (definition)
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.
9 downstream results would collapse with it. See the blast radius on the graph →