End behavior of Ramanujan's taxicab numbers
Number Theory
2024-04-15 v1 Combinatorics
Abstract
Generalized taxicab numbers are the smallest positive integers that are the sum of exactly , positive -th powers in exactly distinct ways. This paper is considers for which values of does a smallest such integer exist as gets large. There appear to be only two possible outcomes, leading to curious results like there is no positive integer that can be expressed as the sum of exactly positive squares in exactly ways. This paper resolves a number of conjectures found in the OEIS by considering generalized Taxicab numbers in the setting of the theory of partitions.
Keywords
Cite
@article{arxiv.2404.08190,
title = {End behavior of Ramanujan's taxicab numbers},
author = {Brennan Benfield and Oliver Lippard and Arindam Roy},
journal= {arXiv preprint arXiv:2404.08190},
year = {2024}
}