Fundamental Theorem of Arithmetic

Intro to Proofs · Axiom Academy

LESSON Fundamental Theorem of Arithmetic Every integer greater than 1 is a product of primes — in exactly one way. We prove both halves: that the factorization always exists, and that it is unique. 1. Existence — Every Number Breaks Into Primes The theorem has two halves. First, existence : every integer n > 1 can be written as a product of primes. The picture is a factor tree — split n into any two smaller factors, then keep splitting until every leaf is prime. A composite splits into two smaller factors… …and the prime leaves are the factorization 2. Uniqueness — All Roads Lead to the Same Primes Existence lets you build a tree some way. But you had choices: 60 splits as , or , or . Uniqueness is the claim that no matter which splits you pick, you always land on the same multiset of primes . and start two visibly different trees. Both bottom out at the primes 2, 2, 3, 5 — same primes, same counts. "Unique" means up to order: and are the same. Collecting equal primes: — the canonical form. The shape of the uniqueness proof Suppose n had two genuinely different factorizations . Take p_1 : it divides the left side, so it divides the right side, . The next step is the key — and it needs a special property of primes. 3. Euclid's Lemma — Why Primes Are Special

This is the written version of the interactive lesson above. See the full Intro to Proofs course.