BalaA363412Valuation

Abstract

Integer quadratic products and Bala’s valuation conjectures.

Definition 1.1 (The paired valuation conjecture).

$$\mathrm{claim}=\left(\forall n\in \mathbb{N},(1\le n)\Rightarrow (\mathrm{padicValInt}\left(3,BalaA363411.b\left(3\cdot n\right)\right)=\mathrm{padicValInt}\left(3,BalaA363411.b\left(3\cdot n-1\right)\right))\right)$$

Formalization. D5/S3/Arith/Congruence/BalaA363412Valuation.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 second clause starts at n=1 so both source indices are nonnegative. The subtraction 3n-1 is natural truncated subtraction. The source sequence is BalaA363411.b.

Theorem 1.2 (The paired three-adic valuations agree).

$$\forall n\in \mathbb{N},(1\le n)\Rightarrow (\mathrm{padicValInt}\left(3,BalaA363411.b\left(3\cdot n\right)\right)=\mathrm{padicValInt}\left(3,BalaA363411.b\left(3\cdot n-1\right)\right))$$

Proof. Machine-checked in Lean as D5/S3/Arith/Congruence/BalaA363412Valuation.result (✓ std3). ∎

Resolves. Problems/bala3634-paired-valuations (proved) by D5/S3/Arith/Congruence/BalaA363412Valuation.result.

Source. Repository-derived.

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

Commentary.

At indices 3n and 3n-1, the reflection congruences express twice the imaginary coordinate as 3n times an integer prime to 3. For n at least one the factor 3n is nonzero. Multiplicativity of padicValInt removes the unit factors; both valuations equal padicValInt 3 (3n).

References

• Truth anchor: D5/S3/Arith/Congruence/BalaA363412Valuation.claim
• Truth anchor: D5/S3/Arith/Congruence/BalaA363412Valuation.result
• Dependency: D5/S3/Arith/Congruence/BalaA363412