Every integer greater than 1 can be written as a product of primes, and that product is unique (up to the order of the factors).

$$60 = 2^2 \times 3 \times 5$$ $$1001 = 7 \times 11 \times 13$$ $$2 = 2$$

There is no other way to express 60 as a product of primes. This is the Fundamental Theorem of Arithmetic — sometimes called unique prime factorisation.

Two claims in one

The theorem makes two separate claims:

1. Existence: Every integer $n > 1$ has at least one prime factorisation. This is easy to see by induction — if $n$ is prime, done. If not, $n = a \times b$ where both $a, b < n$, and we can factor them in turn. The process terminates because the factors keep getting smaller.

2. Uniqueness: There is exactly one such factorisation. This is the deeper claim. It depends on a key property of primes: if $p$ divides $a \times b$, then $p$ divides $a$ or $p$ divides $b$ (or both). This is called Euclid’s lemma, and it doesn’t hold for all numbers — only primes.

Why uniqueness isn’t obvious

Consider the number $12$. We might factor it as: - $2 \times 2 \times 3$ - $2 \times 6$ → but $6 = 2 \times 3$, so this is the same factorisation - $3 \times 4$ → but $4 = 2^2$, so again the same

Every path leads to the same set of prime factors. This feels inevitable — but it’s actually a special property of the integers. There are other number systems (like $\mathbb{Z}[\sqrt{-5}]$, the integers extended with $\sqrt{-5}$) where factorisation is not unique:

$$6 = 2 \times 3 = (1 + \sqrt{-5})(1 - \sqrt{-5})$$

Both factorisations are into irreducible elements, but they’re genuinely different. Unique factorisation is a feature of the integers, not a law of algebra.

The arithmetic fingerprint

Because prime factorisation is unique, it gives every number a fingerprint — a canonical representation. This fingerprint lets us define:

• GCD and LCM in terms of shared prime factors: $\gcd(12, 18) = 2^1 \times 3^1 = 6$ (take the minimum exponent of each prime)
• Divisibility as a partial order: $a | b$ if and only if every prime power in $a$’s factorisation appears in $b$’s with at least as high an exponent
• Multiplicative functions: Functions like the divisor count $d(n)$ or divisor sum $\sigma(n)$ that can be computed prime-by-prime: $\sigma(p^a) = 1 + p + p^2 + \cdots + p^a$

The fundamental theorem is also why the Euler product works — the fact that every integer has a unique prime factorisation is precisely what lets a sum over all integers be rewritten as a product over primes.

Connection to Number Craft

If you’ve played Number Craft, you’ve already used this theorem. Every number in your collection has a prime factorisation that the game shows you. The factorisation is unique — there’s exactly one way to build each number from primes. The game’s crafting system is the fundamental theorem made playable.