Euler's and the Taxi Cab relations and other numbers that can be written twice as sums of two cubed integers

TL;DR

This paper explores the mathematical relationships between Euler's relation, the Taxi-Cab relation, and sums of two cubes, providing general solutions, recursive formulas, and infinite families of such relations.

Contribution

It introduces a unified framework for solutions to sums of two consecutive cubes equaling sums of two other cubes, including recursive and parametric equations.

Findings

Euler's and Taxi-Cab relations are solutions of the same equation.

Infinite families of relations among sums of two cubes are identified.

Explicit recursive and parametric formulas for these relations are derived.

Abstract

We show that Euler's relation and the Taxi-Cab relation are both solutions of the same equation. General solutions of sums of two consecutive cubes equaling the sum of two other cubes are calculated. There is an infinite number of relations to be found among the sums of two consecutive cubes and the sum of two other cubes, in the form of two families. Their recursive and parametric equations are calculated.