Why the FTA is fundamental

Remark — Why it is "fundamental"

The uniqueness of the factorisation is what allows us to compute GCD and LCM, compare divisibility, prove that $\sqrt{2}$ is irrational, and build RSA cryptography. Without uniqueness, arithmetic would become ambiguous: the apparently obvious statement hides a deep theorem. The first complete modern proof is reconstructed starting from the so-called Euclid’s lemma (in the Elements, Book VII, Proposition 30: “if a prime divides a product, it divides one of the factors”).

Many results that we take for granted therefore rest on this theorem: it is the reason why prime factorisation is “fundamental” and not a mere calculation exercise.

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Topics: Numbers and operations
Concepts: Prime factorisation · Fundamental theorem of arithmetic
People: Euclid