Submitted:
08 February 2026
Posted:
10 February 2026
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Abstract
This work rigorously explores the conceptual transition between Pythagorean harmony \( h^2 = a^2 + b^2 \) and Fermatian impossibility \( h^3 = a^3 + b^3 \), explicitly acknowledging that Fermat's Last Theorem (FLT) prohibits integer solutions for \( n=3 \). Starting from Nicomachus' historical formula for the cumulative sum of cubes,
\( S(n) = \sum_{k=1}^n k^3 = \left[\frac{n(n+1)}{2}\right]^2 \),
and applying the first-order retrospective finite difference operator \( \nabla S(n) = S(n) - S(n-1) \), we \( \textbf{deduce} \) the algebraic identity:
\( n^3 = \frac{n^2}{4}\big[(n+1)^2 - (n-1)^2\big] \).
It is crucial to emphasize that Nicomachus (c. 100 CE) \( \textbf{did not explicitly formulate} \) this identity in terms of symmetric differences; his historical contribution was exclusively limited to the cumulative sum formula. The expression above constitutes a \( \textbf{modern deduction} \) derived via discrete calculus. Using this deduction, we construct an exact symbolic representation:
\( h = \sqrt[3]{\frac{a^2}{4}\big[(a+1)^2 - (a-1)^2\big] + \frac{b^2}{4}\big[(b+1)^2 - (b-1)^2\big]} \)
We demonstrate that this expression, while mathematically exact and constructed exclusively through integer operations, does not produce \( h \in \mathbb{Z} \) —empirically confirming the arithmetic obstruction of FLT through 2,500 numerical verifications (\( 1 \leq a,b \leq 50 \)). We establish the combinatorial uniqueness of exponent \( k=2 \) in symmetric differences \( (n+1)^k - (n-1)^k \), revealing why the compact representation works exclusively for cubes. We contextualize historically the problem from the Pythagorean school (6th century BCE) to Wiles' proof (1994), highlighting contributions from Nicomachus, Euler, Sophie Germain, and Kummer with historiographical rigor. The genuine value of this proposal resides in its pedagogical capacity to illustrate the fundamental distinction between \( \textit{internal structure} \) (local properties of individual cubes) and \( \textit{additive structure} \) (relations between distinct cubes), honestly transforming Fermatian impossibility into an opportunity to comprehend the structural limits inherent to mathematics.
