A New Method in the Problem of Three Cubes

TL;DR

This paper introduces a new algorithm for finding integer solutions to the sum of three cubes problem, focusing on polynomial degree minimization and exploring related polynomial equations.

Contribution

It presents a novel method for constructing high-degree polynomials P1, P2, P3 and low-degree Q to solve the sum of three cubes problem.

Findings

Developed an algorithm for specific values of d

Analyzed the relationship between polynomial degrees and solutions

Explored minimization of polynomial degree in solutions

Abstract

In the current paper we are seeking P1(y),P2(y),P3(y) with the highest possible degree polynomials with integer coefficients, and Q(y) via the lowest possible degree polynomial, such that P1(y)^3+P2(y)^3+P3(y)^3=Q(y). Actually, the solution of this problem has close relation with the problem of the sum of three cubes a^3+b^3+c^3=d, since deg Q(y)=0 case coincides with above mentioned problem. It has been considered estimation of possibility of minimization of deg Q(y). As a conclusion, for specific values of d we survey a new algorithm for finding integer solutions of a^3+b^3+c^3=d.