Finite Differences of Prime Powers as Cyclotomic Norms: A Structural Bridge from Nicomachus to Euler. Universal Anderson–Faulhaber–Bernoulli Identity: Internal Structure of Perfect Powers and Arithmetic Obstruction via Discrete Calculus

For every prime p and every integer a, the backward finite difference δp(a) := a^p − (a − 1)^p equals the cyclotomic binary form Φp(a, a − 1), where Φp(X, Y) is the homogenisation of the p-th cyclotomic polynomial, and hence equals the norm NQ(ζp)/Q(a − ζp(a − 1)). For p = 3 this specialises to the identity δ3(a) = NZ[ω](a − ω(a − 1)), where ω = e^(2πi/3), connecting the individual cubic finite difference obtained by differencing the classical sum formula of Nicomachus of Gerasa (~100 CE) with the Eisenstein norm that appears in Euler's factorisation of a^3 + b^3. We develop this identity in three directions: (a) General cyclotomic framework. For each prime p, every prime divisor q of δp(a) satisfies q ≡ 1 (mod p), imposing an arithmetic sieve whose density ~1/(p−1) grows increasingly severe with p. (b) Arithmetic density. The values {δ3(a)}a≥1 form a thin subfamily of the Löschian numbers (norms in Z[ω]), with counting function ~√(N/3) versus the Landau-Ramanujan asymptotic CN/√log N for all Löschian numbers up to N. (c) Three-language equivalence. For the cubic case we prove a precise equivalence among: (i) divisibility of δ3(a), (ii) multiplicative order modulo q, and (iii) splitting of q in Z[ω]. We also give an elementary proof of the base case 1 + b^3 = c^3 (no positive-integer solutions), and derive 3-adic constraints on any hypothetical solution to a^3 + b^3 = c^3 via the Lifting-the-Exponent Lemma, without invoking unique factorisation in Z[ω].