Elementary Formulas for Greatest Common Divisors and Semiprime Factors

TL;DR

This paper introduces new elementary, fixed-length formulas for computing the GCD of two integers and extracting prime factors of semiprimes, using basic arithmetic operations, simplifying previous methods.

Contribution

It presents simplified, elementary formulas for GCD and semiprime factorization derived from existing formulas and novel arithmetic expressions for square roots.

Findings

New fixed-length formulas for GCD using basic operations

Elementary formula for prime factors of semiprimes

Derived from simplified GCD formulas and square root expressions

Abstract

We conjecture new elementary formulas for computing the greatest common divisor (GCD) of two integers, alongside an elementary formula for extracting the prime factors of semiprimes. These formulas are of fixed-length and require only the basic arithmetic operations of: addition, subtraction, multiplication, division with remainder, and exponentiation. Our GCD formulas result from simplifying a formula of Mazzanti and are derived using Kronecker substitution techniques from our earlier research. By applying these GCD formulas together with our recent discovery of an arithmetic expression for $\sqrt{n}$, we are able to derive explicit elementary formulas for the prime factors of a semiprime $n=p q$.