{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:an-erdos-kac-law-for-base-b-palindromes-and-for-reversed-primes",
  "title": "An Erdős–Kac law for base-$b$ palindromes and for reversed primes",
  "canonical_statement": "For every base $b\\ge2$, the number of prime factors of the $\\lambda$-digit base-$b$ palindromes, and of the base-$b$ reversals of the $\\lambda$-digit primes, obeys an Erdős–Kac law: counted with or without multiplicity, it is asymptotically normal with centring $\\log\\log b^{\\lambda}$ and scaling $\\sqrt{\\log\\log b^{\\lambda}}$. For reversed primes the law persists when the leading digit of the prime is prescribed. Erdős–Kac laws were already known for other digitally defined families — integers with a fixed digit sum, or with digits restricted to a fixed set — but for palindromes only the largest value of $\\omega$ had been studied, with nothing known about the typical value, and for reversed primes the level of distribution the argument needs became available only in 2025.",
  "plain_summary": "VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "resolved",
  "solution_events": [
    {
      "id": "vibemathed:an-erdos-kac-law-for-base-b-palindromes-and-for-reversed-primes:event",
      "problem_id": "vibemathed:an-erdos-kac-law-for-base-b-palindromes-and-for-reversed-primes",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-08-01T00:00:00.000Z",
      "title": "Zenodo preprint",
      "summary": "New theorems, not a formalisation of previously known results. For every base $b\\ge2$ the Erdős–Kac law is established for the $\\lambda$-digit base-$b$ palindromes and for the base-$b$ reversals of the $\\lambda$-digit primes, for $\\omega$ and $\\Omega$ and for $\\omega_S,\\Omega_S$ with any regular set $S$ of primes; with normal order $\\log\\log n$ on both families, and, for $\\omega$, all moments of order up to $\\tfrac12(\\log\\log b^{\\lambda})^{1/3}$ uniformly in the order. For reversed primes it also holds with the leading digit prescribed.\n\nNot settled: the results rest on quoted inputs (Col for palindromes, the Bombieri–Vinogradov theorem of Dartyge–Rivat–Swaenepoel for reversed primes), and both families exclude the primes dividing $b(b^{2}-1)$. No rate of convergence is obtained. The question of Banks–Shparlinski on the *largest* value of $\\omega$ on palindromes is untouched: the trivial bound $\\ll\\lambda/\\log\\lambda$ and their $\\lambda^{o(1)}$ remain far apart.",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:an-erdos-kac-law-for-base-b-palindromes-and-for-reversed-primes:attempt"
      ],
      "method_family_ids": [
        "vibemathed:an-erdos-kac-law-for-base-b-palindromes-and-for-reversed-primes:method"
      ],
      "source_assertion_ids": [
        "vibemathed:an-erdos-kac-law-for-base-b-palindromes-and-for-reversed-primes:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:an-erdos-kac-law-for-base-b-palindromes-and-for-reversed-primes:attempt",
      "problem_id": "vibemathed:an-erdos-kac-law-for-base-b-palindromes-and-for-reversed-primes",
      "model": "Claude Fable 5, Claude Opus 5",
      "provider": "Anthropic",
      "attempted_at": "2026-08-01T00:00:00.000Z",
      "human_collaborators": [],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "Used at every stage: literature search, jointly working out the main arguments, and drafting the manuscript. Two contributions were decisive. The arithmetic input for palindromes — Col's theorem on the level of distribution of palindromes in arithmetic progressions — was located by the models. And the general Erdős–Kac criterion used here, a modification of the Granville–Soundararajan sieve moment estimate that allows a finite exceptional set of primes at which the local densities are arbitrary, was worked out jointly with them. The models also produced the Lean 4 formalisation. The author verified all statements, proofs and references, made the final decisions on content and presentation, and is responsible for any remaining errors.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "Lean 4 formalisation, axiom audit and verification report",
          "url": "https://github.com/vibefrtz/vibemath",
          "kind": "code",
          "license": null
        },
        {
          "label": "Archived v1.0.0 (version DOI)",
          "url": "https://doi.org/10.5281/zenodo.22078540",
          "kind": "paper",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "Zenodo preprint",
          "url": "https://doi.org/10.5281/zenodo.22078539",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: An Erdős–Kac law for base-$b$ palindromes and for reversed primes",
          "url": "https://vibemathed.com/problem/an-erdos-kac-law-for-base-b-palindromes-and-for-reversed-primes",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:an-erdos-kac-law-for-base-b-palindromes-and-for-reversed-primes:verification",
      "problem_id": "vibemathed:an-erdos-kac-law-for-base-b-palindromes-and-for-reversed-primes",
      "solution_event_id": "vibemathed:an-erdos-kac-law-for-base-b-palindromes-and-for-reversed-primes:event",
      "level": "lean_checked_statement_unaudited",
      "mathematical_correctness": "supported",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "Read independently here on 24 August 2026 at github.com/vibefrtz/vibemath, in addition to the submission's own detailed VERIFICATION.md, which this confirms rather than repeats. All 21 Lean files (about 7850 lines) carry no sorry, no admit and no native_decide; the sole textual match for \"axiom\" outside Cited.lean is a comment, not a declaration. axiom_audit.txt shows every one of the 27 theorems drawing only Lean's three standard axioms plus a subset of the eight declared in Cited.lean, matching the paper's citation structure theorem by theorem. Spot-checked Main.lean against the manuscript: pal_EK_omega and rev_EK_omega are Tendsto statements of the empirical distribution to Phi(t) with centring and scaling LL b lam and its square root, matching Theorems 1.1 and 1.2 as stated. Of the eight cited results, Col, Banks-Shparlinski, Dartyge-Rivat-Swaenepoel and Granville-Soundararajan were confirmed to exist with the stated venues; Dartyge-Rivat-Swaenepoel (arXiv:2506.21642) is from June 2025, corroborating the submission's claim that the reversed-prime argument's input became available only that year. This is a sampling audit, not the full informal-to-formal correspondence review the lean-verified tier requires, so the tier stays where the submission itself placed it.",
      "sources": [
        {
          "label": "VibeMathed: An Erdős–Kac law for base-$b$ palindromes and for reversed primes",
          "url": "https://vibemathed.com/problem/an-erdos-kac-law-for-base-b-palindromes-and-for-reversed-primes",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "Zenodo preprint",
          "url": "https://doi.org/10.5281/zenodo.22078539",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "Zenodo preprint",
      "url": "https://doi.org/10.5281/zenodo.22078539",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: An Erdős–Kac law for base-$b$ palindromes and for reversed primes",
      "url": "https://vibemathed.com/problem/an-erdos-kac-law-for-base-b-palindromes-and-for-reversed-primes",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:an-erdos-kac-law-for-base-b-palindromes-and-for-reversed-primes:signal:replay_ready",
      "problem_id": "vibemathed:an-erdos-kac-law-for-base-b-palindromes-and-for-reversed-primes",
      "kind": "replay_ready",
      "reason": "A public source and a non-unreviewed verification label support replay triage.",
      "generated_at": "2026-09-13T16:28:40.178Z"
    }
  ],
  "uncertainties": [
    "VibeMath has not independently verified the mathematical claim.",
    "AI-attempt independence and training-data exposure are unknown unless explicitly documented.",
    "VibeMath has not independently audited the mathematical statement, proof, or novelty claim."
  ],
  "generated_at": "2026-09-13T16:28:40.178Z"
}