A Graph-Theoretic Proof of the Fundamental Theorem of Arithmetic with Algorithmic Construction

The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be factored uniquely into prime numbers, up to order. In this paper, we present a graph-theoretic framework to understand and prove this theorem. By modeling factorization as a tree structure, we define factorization trees and establish four supporting lemmas. We then develop a recursive algorithm for constructing these trees. This approach not only yields a constructive proof of the theorem but also offers visual and algorithmic insights into the uniqueness of prime factorizations.