An Erdős–Kac law for base-$b$ palindromes and for reversed primes
resolvedconfidence 70%
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Precise 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.
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fidelity, correctness, priority, or novelty.
What AI did
Claude Fable 5, Claude Opus 5
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.
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.
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.
Not 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.
Known method families
argument (source-reported)
Source-reported tools: argument.
Independent: unknown · difference confidence: 0
What remains uncertain
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AI-attempt independence and training-data exposure are unknown unless explicitly documented.
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.