Date of Award

5-2026

Document Type

Thesis

Degree Name

Master of Arts in Mathematics

Department

Mathematics

First Reader/Committee Chair

Fejzic, Hajrudin

Abstract

This thesis investigates extensions and structural generalizations of the classical identity \[ \sum_{k=1}^{n} k^3 = \left( \sum_{k=1}^{n} k \right)^2, \] traditionally attributed to Nicomachus of Gerasa. Despite its simple look, this cube - square identity reveals connections between combinatorics, multiplicative number theory, and Diophantine equations.

We begin by presenting an expanded combinatorial proof of the identity based on Stein’s rectangle - counting argument, clarifying the geometric structure underlying the formula. We then establish a multiplicative analogue using Euler’s divisor-counting function \( \tau(n) \), proving that \[ \sum_{d \mid n} \tau(d)^3 = \left( \sum_{d \mid n} \tau(d) \right)^2, \] thereby extending the cube - square phenomenon from consecutive integers to divisor sums.

Next, we study a modified cubic identity obtained by replacing one cube \(k^3\) with another cube \(x^3\). We obtain a complete parametric classification of all triples \((k,x,n)\) for which the modified identity holds, including a uniform family discovered by students. We generalize an observation of Paul Loomis concerning identities of Nicomachus type and provide a substantially simplified proof that repetition of terms is necessary in all nonconsecutive cases. We further show that the parametrization of solutions is shown by a Pell equation, \[ X^2 - 3Y^2 = 1, \] and derive explicit recurrence relations for the associated solution sequences. This gives us an unexpected bridge between additive cube–square identities and classical Diophantine theory.

Finally, we determine arithmetic restrictions on admissible values of \(n\), showing that the existence of nontrivial solutions is governed by the prime factorization of the quadratic polynomial \(n^2+n+1\).

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