slug: bala3634-type-one
bibkey: bala2023sqrtminusthree
doi: null
url: https://oeis.org/A363411
triage: theorem
motivation_gids:
  - D5/S3/Arith/Congruence/BalaA363411TypeOne.result

Type-one primes through 101

Problem

$\forall p\le 101$, the primes dividing no $a(n)$ are exactly $\{3,11,17,23,29,41,47,53,59,83,89,101\}$, with the term index $n$ unbounded. Here $a(n)$ and $b(n)$ are the integer coordinates of ${\prod }_{k=0}^n(1+k\sqrt{-3})$. Preregistration #18634 maps to D5.S3.Arith.Congruence.BalaA363411TypeOne.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 #18634 found no proof of this exact assertion in the searched scope; completeness is ASSUMED-UNVERIFIED.

Route

BalaA363411TypeOne.nonzero_of_block extends each nonzero residue block to all indices by strong induction. BalaA363411.real_one handles the prime $3$; listed_data and unlisted_data certify the exact finite prime list. 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.BalaA363411TypeOne.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.