Representation Of Integers By Sum Of Three Cubes, A New Approach Based On Seed Equation

TL;DR

This paper introduces a novel algebraic approach using seed equations to represent integers as sums of three cubes, enabling the derivation of multiple solutions and parametrizations through transforming seed equations into quadratic and linear forms.

Contribution

The paper presents a new algebraic method based on seed equations for expressing integers as sums of three cubes, including multiple solutions and parametrizations, with alternative methods for difficult cases.

Findings

Multiple solutions for sum of three cubes identified

Parametrization formulas for solutions provided

New algebraic approach simplifies solving the problem

Abstract

We have proved in this paper that numbers can be expressed in algebraic form using one variable and two real rational quantities and thus sum of three cubes can also be expressed in algebraic form as a cubic polynomial. Using skeletal or seed equation, the polynomial can be transformed into a quadratic equation. A seed equation denotes a simple equation that represents a given integer as sum of three cubes including 0, for example, a seed equation for integer 2 is 1 cubed plus 1 cubed plus 0 cubed. Resultant quadratic equation can further be transformed into a linear equation which yields value of the variable and substitution of the value of the variable into the algebraic form of numbers results in the required solution. Notwithstanding, finding a single set of three cubes, we have found, using this approach, multiple sets of cubes. We have also given parametrisation for such cubes. Cases, where it is difficult or not possible to determine a seed equation, alternate method has been provided. Methods used are unattempted, innovative and easily comprehensible.