On Hunting for Taxicab Numbers

TL;DR

This paper explores properties of numbers expressible as sums of two equal odd powers, developing algorithms to find new taxicab numbers with multiple representations as sums of positive cubes, extending previous computational results.

Contribution

It introduces a novel algorithmic approach to identify taxicab numbers with up to 14 representations, expanding the known set of such numbers.

Findings

Computed new taxicab numbers with 7 to 14 representations

Developed methods for analyzing diophantine equations related to taxicab numbers

Enhanced algorithms for verifying minimality of taxicab numbers

Abstract

In this article, we make use of some known method to investigate some properties of the numbers represented as sums of two equal odd powers, i.e., the equation $x^n+y^n=N$ for $n\ge3$. It was originated in developing algorithms to search new taxicab numbers (i.e., naturals that can be represented as a sum of positive cubes in many different ways) and to verify their minimality. We discuss properties of diophantine equations that can be used for our investigations. This techniques is applied to develop an algorithm allowing us to compute new taxicab numbers (i.e., numbers represented as sums of two positive cubes in $k$ different ways), for $k=7...14$.