From the Pythagorean Dream to the Fermatian Obstruction: Symbolic Representation of h=p√(ap+bp ) via an Identity Derived from Nicomachus’ Cumulative Sum

This paper presents, in a unified form and with explicit logical justification, a chain of original results on the internal structure of perfect powers and their connection with Fermat’s Last Theorem (FLT). Starting from the historical formula of Nicomachus of Gerasa c. 100 AD for the cumulative sum of cubes, ( S_3(n) = T_n^2 ), and applying the backward finite-difference operator ∇—formalised by Taylor (1715) and systematised by Boole (1860)—the the Anderson Identity (2026) is derived:
( n^3 = frac{n^2}{4}bigl[(n+1)^2-(n-1)^2bigr] = T_n^2 – T_{n-1}^2. )
This identity, valid exclusively for cubes, is extended to every ( pge1 ) through the Universal Anderson–Faulhaber–Bernoulli Identity (2026):
( n^p = frac{1}{p+1}sum_{j=0}^{p}binom{p+1}{j}B_j^{+},delta_{p+1-j}(n),
qquad delta_m(n):=n^m-(n-1)^m, ) (UI)
derived by applying ( nabla ) to the classical Faulhaber–Bernoulli formula for cumulative sums of powers. The quantity ( delta_m(n)=n^m-(n-1)^m )—the finite difference of the individual m-th power—constitutes the original internal perspective of this work: it reorients the historical cumulative-sum formula toward individual powers, revealing that the internal algebraic complexity of ( n^p ) grows as ( C(p)=lfloor p/2rfloor+1 ), with ( p=3 ) as the unique point of optimal compactness (pure monomial).The second contribution is Universal Symbolic Representation (2026):
( h=sqrt[p]{I_p(a)+I_p(b)},quad hnotinmathbb{Z} forall,pge3, ) (SR)
which expresses ( h=sqrt[p]{a^p+b^p} ) through purely integer operations in the radicand and establishes that its irrationality for ( pge3 ) is a structurally inevitable consequence, not an accidental one. Full step-by-step derivations, explicit expansions for ( p=2,ldots,8 ), 50-digit-precision numerical verifications for 10,000 pairs ( (a,b) ) with ( 1le ale ble100 ), and the conceptual gradation of the Fermatian obstruction in three regimes—quadratic, cubic, and Bernoulli—are presented. The Structural Stratification Theorem is proved: ( C(p)=lfloor p/2rfloor+1 ), with ( p=3 ) as the unique point of optimal compactness. Complete chronological historical contextualisation from the Pythagoreans to Wiles, analysis of the originality of the present perspective, generalised symmetry breaking, and genuine pedagogical value are included.