BalaA363412

Abstract

Integer quadratic products and Bala’s valuation conjectures.

Definition 1.1 (Nondivisibility on the third residue class).

$$\mathrm{claim}=\left(\forall n\in \mathbb{N},\neg ((3:\mathbb{Z})\mid BalaA363411.b\left(3\cdot n+1\right))\right)$$

Formalization. D5/S3/Arith/Congruence/BalaA363412.claim (✓ std3).

Citation. P. Bala (2023). OEIS A363411 and A363412 — quadratic integer product coordinates. URL: https://oeis.org/A363411.

Commentary.

OEIS A363412, FORMULA (revision 19, no page): “Conjectures: 3 does not divide a(3n+1) for all n; the 3-adic valuation v_3(a(3n)) = v_3(a(3*n-1)) for all n.” The first clause concerns every natural n, including zero; the source sequence is BalaA363411.b.

Theorem 1.2 (Three never divides these terms).

$$\forall n\in \mathbb{N},\neg ((3:\mathbb{Z})\mid BalaA363411.b\left(3\cdot n+1\right))$$

Proof. Machine-checked in Lean as D5/S3/Arith/Congruence/BalaA363412.result (✓ std3). ∎

Resolves. Problems/bala3634-nondivisibility (proved) by D5/S3/Arith/Congruence/BalaA363412.result.

Source. Repository-derived.

Acknowledgement. P. Bala (2023). OEIS A363411 and A363412 — quadratic integer product coordinates. URL: https://oeis.org/A363411.

Commentary.

The coordinate recursion modulo 3 has period 3. At 3n+1 its value is (1,1), so the imaginary coordinate is nonzero modulo 3.

Theorem 1.3 (Two integer reflection congruences).

$$\forall m\in \mathbb{N},(3\cdot (m:\mathbb{Z})\mid 2\cdot BalaA363411.b\left(m\right)-(m:\mathbb{Z})\cdot \left((m:\mathbb{Z})+1\right))\wedge (3\cdot \left((m:\mathbb{Z})+1\right)\mid 2\cdot BalaA363411.b\left(m\right)-(m:\mathbb{Z})\cdot \left((m:\mathbb{Z})+1\right))$$

Proof. Machine-checked in Lean as D5/S3/Arith/Congruence/BalaA363412.reflection_congruences (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. P. Bala (2023). OEIS A363411 and A363412 — quadratic integer product coordinates. URL: https://oeis.org/A363411.

Commentary.

For every finite integer coefficient family, a uniform shift by c changes the real coordinate by a multiple of 3c and the imaginary coordinate by c times the cardinality modulo 3c. Apply this to k mapped to m-k on 0 through m, and to m+1-k on 1 through m. Conjugation reverses the imaginary coordinate. The two displayed divisibilities follow and determine the paired three-adic valuations.

References

• Truth anchor: D5/S3/Arith/Congruence/BalaA363412.claim
• Truth anchor: D5/S3/Arith/Congruence/BalaA363412.reflection_congruences
• Truth anchor: D5/S3/Arith/Congruence/BalaA363412.result
• Dependency: D5/S3/Arith/Congruence/BalaA363411