Golden Cubic Blocks Are Not Fifth Powers
Abstract
Every actual Lucas cubic block is not a fifth power by an induction over every positive depth.
Theorem 1.1 (No golden cubic block is an integer fifth power).
Proof. Machine-checked in Lean as D5/S3/Factorization/GoldenCubicBlockNonFifthPower.golden_cubic_block_not_fifth_power (✓ std3). ∎
Citation. Yann Bugeaud, Maurice Mignotte, Samir Siksek (2006). Classical and modular approaches to exponential Diophantine equations II. The Lebesgue-Nagell equation. DOI: 10.1112/S0010437X05001739.
Commentary.
The integer exclusion is a specialization of the classical D=3 Lebesgue-Nagell result recorded in the cited literature: x^2+3 is not an integer fifth power. Substitute x=L_(3^j). The repository formal proof uses the following modular argument, with no novelty claim. The convention is L_n = trace(phi^n), with L_1 = 1. The existing actual Lucas recurrence sends x to x(x^2+3) at power-of-three indices. Induction over all j >= 1 gives the orbit 4, 10, 7, 1 modulo 11. The block has residue 8 or 4, whereas fifth powers have residue 0, 1, or 10. The finite checks establish the transition and obstruction; the theorem ranges over every positive depth and every integer, including even depths. This supplies the exponent-five part of DCE.1-DCE.2 and excludes the single-prime fifth-power alternative in DCE.3. It does not supply their arbitrary-exponent exclusion or close those parent obligations. For the original natural block B_j = natAbs(L_(3^j)^2+3), nonnegativity identifies its integer cast with this radicand. If a rational q satisfies q^5 = B_j, it is integral over the integers, since its fifth power is an integer. Integral closedness gives an integer q, contradicting the theorem. The prime-exponent Kummer polynomial criterion at exponent 5 then gives irreducibility of X^5-C(B_j) over the rationals. These are applications of this theorem and the existing integrality and irreducibility APIs. They do not establish a compositum degree, an earlier-prime splitting condition, a Frobenius class, a density statement or a Wall-Sun-Sun conclusion.