On the Equation A³+B³=Cⁿ: Elementary Methods, Modular Obstructions, and a Brauer-Manin Research Program

Abstract: We study primitive integer solutions to A³+B³=Cⁿ with gcd (A, B, C) =1 and n≥3. Novel results include: a p-adic valuation closure of Case 2 (gcd (A+B, A²-AB+B²) =3) for all n≥3 via a Ljunggren-type obstruction; explicit vector composition in Zω formalizing composite exponent reduction; a synthesis closing all split Cartan obstruction primes (n≡1 mod 3, n>13) via Bilu-Parent-Rebolledo combined with an explicit j-invariant argument; closure of non-split Cartan primes above 1. 4×10⁷ via Le Fourn-Lemos and the observation that j (E₀, ₁) is never an integer for primitive solutions. The remaining gap — approximately 169, 000 non-split Cartan primes between 13 and 1. 4×10⁷ — is precisely identified. We propose a Brauer-Manin research program on the fiber product Zₙ = X₀ (3) ×X (1) Xns^+ (n) as a path to closing it, and establish a conditional proof assuming ABC. Repository disclaimer: verify all Galois-modular hypotheses against primary literature before citation as settled mathematics.