Golden Cubic Blocks Are Not Cubes

Abstract

Every actual Lucas cubic block is a noncube by a finite residue obstruction.

Theorem 1.1 (No golden cubic block is an integer cube).

$$\forall j\in \mathbb{N},1\le j\Rightarrow \neg \exists t\in \mathbb{Z},t^3={\left(L_{3^j}\right)}^2+3$$

Proof. Machine-checked in Lean as D5/S3/Factorization/GoldenCubicBlockNoncube.golden_cubic_block_not_cube (✓ std3). ∎

Source. Repository-derived.

Commentary.

The cubic Lucas recurrence cycles through 4, 6, 3, 1 modulo 7. The block therefore has residue 5 or 4 modulo 7, while integer cubes have residue 0, 1, or 6. The obstruction applies at every positive power-of-three index.

References

• Truth anchor: D5/S3/Factorization/GoldenCubicBlockNoncube.golden_cubic_block_not_cube
• Dependency: D5/S1/Scale/GoldenCubicBlockCongruences