 This lack of unique factorization is a major difficulty for solving Diophantine equations.For example, many wrong proofs of Fermat's Last Theorem (probably including Fermat's ""truly marvelous proof of this, which this margin is too narrow to contain"") were based on the implicit supposition of unique factorization.This difficulty was resolved by Dedekind, who proved that the rings of algebraic integers have unique factorization of ideals