Translations and extensions of the Nicomachus identity

TL;DR

This paper explores generalizations of the Nicomachus identity through parameter translations, sequence modifications, and geometric interpretations, leading to new identities and insights into quadratic forms and cubic equations.

Contribution

It introduces novel generalizations of the Nicomachus identity by incorporating translation and sequence parameters, expanding the scope of known identities.

Findings

Discovery of many new Nicomachus-type identities.

Connection between identities and solutions of quadratic forms.

Geometric interpretation of sequence extensions.

Abstract

We search for Nicomachean identities by adding translation parameters, variable parameters, sequence products and adjoining further numbers to sequences. The solutions of definite and indefinite quadratic forms arise in this study of cubic equations obtained from translation parameters. Our search leads to many general Nicomachean-type identities. We also study the geometry of adjoining two numbers to sequences satisfying the Nicomachean identity.