From Cumulative Sum to Finite Difference: Nicomachus’ Cubic Identity as a Manifestation of Discrete Calculus

This work establishes a rigorous structural connection between Nicomachus' classical formula for the cumulative sum of cubes, \( S(n) = \sum_{k=1}^{n} k^3 = \frac{n^2 (n+1)^2}{4} \), and the algebraic identity: \( n^3 = \frac{n^2}{4}\big[(n+1)^2 - (n-1)^2\big] \), through the first-order finite difference operator nabla \( \nabla S(n) = S(n) - S(n-1) \). We demonstrate that this identity constitutes the discrete manifestation of the fundamental theorem of calculus applied to the quartic sequence \( S(n) \sim n^4/4 \), revealing that cubes emerge as "discrete derivatives" of a quartic polynomial function. We establish the combinatorial uniqueness of the case \( k=2 \) in the symmetric difference \( (n+1)^k - (n-1)^k \), a phenomenon that explains the elegance of the compact representation for cubes. We present exhaustive numerical verifications for \( n = 1 \) through \( n = 25 \), analysis of the expression \( h = \sqrt[3]{a^3 + b^3} \) for pairs \( (a,b) \) with \( 1 \leq a,b \leq 50 \), and historical connections with Pythagorean figurate arithmetic, Boole's umbral calculus, and Faulhaber's theorem. The work highlights the pedagogical value of this perspective for understanding the conceptual transition between classical arithmetic and modern discrete analysis, illustrating the fundamental distinction between internal properties (structure of individual powers) and additive properties (relations between distinct powers), without misrepresenting the theoretical scope of the presented identity.