slug: bala3634-paired-valuations
bibkey: bala2023sqrtminusthree
doi: null
url: https://oeis.org/A363412
triage: theorem
motivation_gids:
  - D5/S3/Arith/Congruence/BalaA363412Valuation.result

Paired three-adic valuations

Problem

$v_3(b(3n))=v_3(b(3n-1))$ for every $n\in \mathbb{N}$ with $n\ge 1$. Here $a(n)$ and $b(n)$ are the integer coordinates of ${\prod }_{k=0}^n(1+k\sqrt{-3})$. Preregistration #18636 maps to D5.S3.Arith.Congruence.BalaA363412Valuation.result (Proved).

Motivation

This is an explicit OEIS conjecture in the source family documented in bala3634-valuations.md.

Gap

The source and literature screen recorded in #18636 found no proof of this exact assertion in the searched scope; completeness is ASSUMED-UNVERIFIED.

Route

BalaA363412.reflection_congruences states that $2b(m)-m(m+1)$ is divisible by both $3m$ and $3(m+1)$. At $m=3n$ and $m=3n-1$, BalaA363412Valuation.at_multiple and before_multiple convert these divisibilities into factorizations of twice each imaginary coordinate as $3n$ times an integer prime to $3$. Their helper BalaA363412Valuation.double_factor applies padicValInt.mul to obtain valuation $v_3(3n)$ on both sides. The primary dossier gives the four issue-to-result mappings.

Falsifier

An index or prime within the displayed domain violating the assertion would refute it.

Evidence

The kernel-checked declaration is D5.S3.Arith.Congruence.BalaA363412Valuation.result. The axiom closure of every public declaration is contained in {propext, Classical.choice, Quot.sound}.

Triage

What the settlement shows

The conclusion holds throughout its stated domain. The primary dossier gives the shared mechanism and distinguishes the finite prime bound from the unbounded term indices. Broader classifications remain open there.

ASSUMED-UNVERIFIED

Literature completeness and priority are ASSUMED-UNVERIFIED. Four-slot escape registration is unfinished in #21978; no declared_validated status is asserted.