A Closed-Form Symbolic Generator: $A^n + B^n = C^n + D^n$, for $n = 2,3$

TL;DR

This paper introduces a novel closed-form symbolic generator for integer solutions to the equation A^n + B^n = C^n + D^n for n=2 and 3, providing explicit formulas and a recursive parametrization leveraging radicals and algebraic numbers.

Contribution

It presents the first symbolic, recursive generator for solutions to the equation for n=2 and 3, using algebraic parametrization and nested radicals, advancing the understanding of such Diophantine equations.

Findings

Explicit formulas for n=2 solutions via differences of squares.

A recursive symbolic generator for n=3 solutions including the Hardy-Ramanujan number.

Provides a closed-form parametrization using radicals and algebraic numbers.

Abstract

We present a unified framework for constructing integer solutions to $A^{n} + B^{n} = C^{n} + D^{n}$ for $n=2,3$. For $n=2$, we derive explicit formulas for any solutions via differences of squares. For $n=3$, we introduce general formulas that include the Hardy-Ramanujan number 1729 for instance, we also construct a symbolic generator that produces infinitely many integer solutions to the Diophantine equation A^3 + B^3 = C^3 + D^3 . While the resulting formulas for $A,B,C,D$ from the symbolic generator developed do not span every single number expressible as a sum of two positive cubes in at least two distinct ways, our method provides a closed-form, algebraic parametrization in terms of a single variable, expressing each term as a radical-exponential function of an integer parameter $c_1$. The generator leverages nested radicals and exponents of algebraic numbers, $\alpha, \beta$ derived from the recurrence structure of the Diophantine constraint. This work represents the first symbolic, recursive generator of its kind and offers a pathway toward approaching higher powers of this problem from a different lens. These methods exploit structural links between binomial expansions and Diophantine constraints, offering a foundation for extensions to higher powers.