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The two-prime cubic Thue descent

Abstract

An actual two-prime block gives a primary Eisenstein factorization, a cubic Thue fiber, and a real cyclic cubic field.

Theorem 1.1 (Exact descent for a two-prime Lucas block).

Lean statement: D5/S3/Factorization/QuadraticIdeals/GoldenCubicThueDescent.golden_cubic_exact_thue_descent

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

Source. Repository-derived.

Commentary.

For j at least one and distinct primes P and Q, suppose the block norm of goldenLucas(3^j)-1 is P squared times Q cubed. Then there are Eisenstein integers pi and gamma with norms P and Q. The ideal of pi is prime, and pi is coprime to its conjugate. The principal ideal of pi is the sum of the oriented ideal of the block and the ideal generated by P; the principal ideal of gamma is the corresponding sum with Q. Both factors are one modulo three, and the oriented factor of the block is exactly pi squared times gamma cubed. Their Fibonacci rank valuations are respectively two and three.

Write (a,b) for the integer coordinates of pi and (u,v) for those of gamma, with A = a squared minus b squared and D = 2ab minus b squared. The homogeneous cubic A u cubed - 3D u squared v + 3(D-A)u v squared + A v cubed equals -2. The second coordinate identity is D(u cubed - 3u v squared + v cubed) + (A-D)(3u v(u-v)) = goldenLucas(3^j)-1. The norm identity u squared - uv + v squared = Q holds for these same u and v.

The rational cubic with coefficients (A, -3D, 3(D-A), A) is irreducible. Its splitting field has degree three over Q, cyclic Galois group, and is totally real. The integer cubic has discriminant 81 P^4. For the fixed pi, the set of all integer pairs solving the displayed cubic equation with right side -2 is finite.

Independently of the forward assumption, any integers a,b,u,v with a squared minus ab plus b squared equal to P and u squared minus uv plus v squared equal to Q, the same cubic equation equal to -2, and the second coordinate expression plus one equal to goldenLucas(3^j), satisfy the natural-number identity blockNorm((goldenLucas(3^j)-1).toNat) = P^2 Q^3. This direction does not assume a factorization of the block or any field or primarity hypothesis.

References