Open Journal of Mathematical Sciences
Vol. 7 (2023), Issue 1, pp. 25-30
ISSN: 2523-0212 (Online) 2616-4906 (Print)
DOI: 10.30538/oms2024.0222
Euler’s and the taxi cab relations and other numbers that can be written
twice as sums of two cubed integers
Vladimir PLETSER\(^{1,*}\)
\(^{1}\) European Space Agency (ret.); Pletservladimir@gmail.com
Copyright © 2024 Vladimir PLETSER. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Received: March 4, 2022 – Accepted: April 06, 2024 – Published: May 15, 2024
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.
Keywords:
Sums of two consecutive cubes ; Equal sums of two cubes ; Taxi-Cab
number ; Euler’s relation
number ; Euler’s relation