{
  "schema_version": "1.0.0",
  "source": "vibemath",
  "problem_id": "vibemathed:prime-values-of-digital-functions-along-the-primes",
  "title": "Prime values of digital functions along the primes",
  "canonical_statement": "Every integer-valued strongly b-additive function g with gcd(g(1),…,g(b−1)) = 1 and nonnegative digit mean takes prime values at infinitely many primes, with a Mertens-type formula and normal-order results; the running example resolves the infinitude of OEIS A052034 (De Geest, 1999): infinitely many primes have a prime sum of squared decimal digits.",
  "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:prime-values-of-digital-functions-along-the-primes:event",
      "problem_id": "vibemathed:prime-values-of-digital-functions-along-the-primes",
      "type": "proved",
      "status": "resolved",
      "occurred_at": "2026-08-01T00:00:00.000Z",
      "title": "Zenodo preprint",
      "summary": "For every integer-valued strongly $b$-additive $g$ with $\\gcd(g(1),\\dots,g(b-1))=1$ and digit mean $\\mu_g\\ge0$: $g(p)$ is prime for infinitely many primes $p$. For $\\mu_g>0$, $\\sum 1/p$ over $p<X$ with $g(p)$ prime is $(d_g/\\varphi(d_g))\\log_3X + C_{g,1} + O(1/\\log\\log X)$, likewise for the first $j$ iterates. Also $\\#\\{p\\le x: g(p)\\text{ prime}\\}\\ll\\pi(x)/\\log\\log x$, of that exact order on a large set of $x$, and $\\omega(g(p))$ has normal order $\\log_3 p$.\n\nWhat is new and what is not. For the digit sum $g=s_b$, Harman (2012) already proved both the infinitude and a Mertens formula; the new information there is the remainder tending to a limit rather than being $O(1)$, and the iterated version for $g=s$ is, in the paper's words, \"contained, in a stronger and quantitative form, in Harman\". The new content is the generalization to every such $g$, which delivers the running example $g=S$, the sum of squared decimal digits, and so the infinitude of OEIS A052034.",
      "ai_contribution": "ai_co_developed",
      "attempt_ids": [
        "vibemathed:prime-values-of-digital-functions-along-the-primes:attempt"
      ],
      "method_family_ids": [
        "vibemathed:prime-values-of-digital-functions-along-the-primes:method"
      ],
      "source_assertion_ids": [
        "vibemathed:prime-values-of-digital-functions-along-the-primes:assertion"
      ]
    }
  ],
  "attempts": [
    {
      "id": "vibemathed:prime-values-of-digital-functions-along-the-primes:attempt",
      "problem_id": "vibemathed:prime-values-of-digital-functions-along-the-primes",
      "model": "Claude Opus 5, Claude Fable 5",
      "provider": "Anthropic",
      "attempted_at": "2026-08-01T00:00:00.000Z",
      "human_collaborators": [],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "Wrote most of the manuscript and the entire Lean 4 formalisation under the author's direction; the exposition was afterwards revised by the author and the same models. Numerical checks computed by machine; scripts distributed with the paper.",
      "failed_routes": [],
      "artifacts": [
        {
          "label": "Lean 4 formalisation, axiom audit and verification report",
          "url": "https://github.com/vibefrtz/vibemath",
          "kind": "lean-proof",
          "license": null
        }
      ],
      "sources": [
        {
          "label": "Zenodo preprint",
          "url": "https://doi.org/10.5281/zenodo.22093372",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Prime values of digital functions along the primes",
          "url": "https://vibemathed.com/problem/prime-values-of-digital-functions-along-the-primes",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        }
      ],
      "cost_usd": null,
      "independence": "unknown"
    }
  ],
  "verifications": [
    {
      "id": "vibemathed:prime-values-of-digital-functions-along-the-primes:verification",
      "problem_id": "vibemathed:prime-values-of-digital-functions-along-the-primes",
      "solution_event_id": "vibemathed:prime-values-of-digital-functions-along-the-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": "Lean-checked, and audited here on 26 August 2026 rather than taken on trust. Sixteen files, about 1550 lines, Lean 4.33.0: no sorry, no admit, no native_decide, and exactly one axiom declaration, confined to DigSq/Cited.lean as claimed. The committed axiom_audit.txt matches its own summary exactly - counted here as 32 results resting on Lean's three built-in axioms alone and 8 resting on those plus `mmr`, 40 in all, with the headline A052034_infinite in the second group. Cited.lean quotes Théorème 1 of Martin-Mauduit-Rivat in French verbatim and carries a quantifier-order note explaining that the encoding must be $\\forall\\varepsilon\\,\\exists C\\,\\forall x$, since $\\forall x\\,\\exists C$ would make the axiom vacuous; that reasoning is correct and is the right thing to have worried about. The cited source is real (J. Inst. Math. Jussieu 18 (2019), 189-224) and the preprint the audit compares against resolves.\n\nThree limits, two of them volunteered by the repository itself. Only phases 1-2 are formalised: the Mertens formula, the counting bounds and the normal order are not. The axiom was compared against the preprint, not the paywalled published text. And the source was read by a model rather than a human, with the audit noting that its own §5 \"exists because the first such reading was wrong\". Lean was not compiled here, and axiom_audit.txt is labelled expected output rather than a captured transcript.",
      "sources": [
        {
          "label": "VibeMathed: Prime values of digital functions along the primes",
          "url": "https://vibemathed.com/problem/prime-values-of-digital-functions-along-the-primes",
          "kind": "source-assertion",
          "license": "https://vibemathed.com/data-license"
        },
        {
          "label": "Zenodo preprint",
          "url": "https://doi.org/10.5281/zenodo.22093372",
          "kind": "primary-mathematical-source",
          "license": null
        }
      ]
    }
  ],
  "sources": [
    {
      "label": "Zenodo preprint",
      "url": "https://doi.org/10.5281/zenodo.22093372",
      "kind": "primary-mathematical-source",
      "license": null
    },
    {
      "label": "VibeMathed: Prime values of digital functions along the primes",
      "url": "https://vibemathed.com/problem/prime-values-of-digital-functions-along-the-primes",
      "kind": "source-assertion",
      "license": "https://vibemathed.com/data-license"
    }
  ],
  "recommended_mode": "verify",
  "recommended_exposure": "result_only",
  "opportunity_signals": [
    {
      "id": "vibemathed:prime-values-of-digital-functions-along-the-primes:signal:replay_ready",
      "problem_id": "vibemathed:prime-values-of-digital-functions-along-the-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"
}