Coprime Conjugate Factors of Cubic Lucas Blocks

Abstract

The oriented Eisenstein factor of each cubic Lucas block is coprime to its conjugate.

Theorem 1.1 (The oriented factor and its conjugate generate the unit ideal).

$$\forall j\in \mathbb{N},1\le j⟹\mathrm{IsCoprime}\left(\mathrm{IdealSpan}\left(⟨-2,\mathrm{goldenLucas}\left(3^j\right)-1⟩\right),\mathrm{IdealSpan}\left(\mathrm{star}\left(⟨-2,\mathrm{goldenLucas}\left(3^j\right)-1⟩\right)\right)\right).$$

Proof. Machine-checked in Lean as D5/S3/Factorization/QuadraticIdeals/GoldenEisensteinCoprime.golden_eisenstein_conjugate_coprime (✓ std3). ∎

Source. Repository-derived.

Commentary.

For j at least one, let x be the Lucas number at index 3^j and let eta = -2 + (x - 1) omega in the Eisenstein order, where omega squared plus omega plus one is zero. The principal ideals of eta and its conjugate generate the unit ideal.

The Lucas residue x = 4 modulo 72 gives x = 4 + 72k for an integer k. The norm of eta is x squared plus three, while eta plus its conjugate equals -(x + 3). Since x squared plus three equals (x + 3)(x - 3) + 12 and x + 3 = 7 + 72k, explicit Bezout coefficients make these two integers coprime. Their image in the Eisenstein order supplies a Bezout identity for eta and its conjugate.

References

• Truth anchor: D5/S3/Factorization/QuadraticIdeals/GoldenEisensteinCoprime.golden_eisenstein_conjugate_coprime
• Dependency: D5/S1/Scale/GoldenCubicBlockCongruences
• Dependency: D5/S3/Factorization/QuadraticIdeals/EisensteinOddQuotient