Steepest Ascent Hill Climbing For A Mathematical Problem

TL;DR

This paper introduces a steepest ascent hill climbing algorithm with a novel tree-based approach to find numerical solutions for Diophantine equations, demonstrating effectiveness on complex cases.

Contribution

It presents a new hill climbing method with a tree representation and heuristic for solving Diophantine equations, addressing the lack of general solution methods.

Findings

Successfully finds solutions for equations with large powers

Effective for equations with many variables

Proves the heuristic approach accelerates solution discovery

Abstract

The paper proposes artificial intelligence technique called hill climbing to find numerical solutions of Diophantine Equations. Such equations are important as they have many applications in fields like public key cryptography, integer factorization, algebraic curves, projective curves and data dependency in super computers. Importantly, it has been proved that there is no general method to find solutions of such equations. This paper is an attempt to find numerical solutions of Diophantine equations using steepest ascent version of Hill Climbing. The method, which uses tree representation to depict possible solutions of Diophantine equations, adopts a novel methodology to generate successors. The heuristic function used help to make the process of finding solution as a minimization process. The work illustrates the effectiveness of the proposed methodology using a class of Diophantine equations given by a1. x1 p1 + a2. x2 p2 + ...... + an . xn pn = N where ai and N are integers. The experimental results validate that the procedure proposed is successful in finding solutions of Diophantine Equations with sufficiently large powers and large number of variables.