Fetoh algorithm

This paper introduces a new algebraic, closed-form approach for computing the greatest common divisor (GCD) of two numbers without relying on iterative division. The proposed method, referred to as the Fetoh Algorithm, reformulates the GCD as a proportional algebraic identity based on multiplication, division, and square-root operations. Unlike the classical Euclidean algorithm, the approach extends naturally beyond integers to rational, irrational, and complex numbers while remaining algebraically consistent. The paper provides a formal derivation of the identity, illustrative examples across multiple numerical domains, and a theoretical time complexity analysis, highlighting the algorithm’s non-iterative structure and its potential relevance to symbolic computation and analytic number theory.