What was already known
Bailey and Crandall proved normality for coprime b,c and asked whether mutual nondivisibility suffices. Their exponential-sum estimate supplies the positive direction here; Kano had already covered some noncoprime examples, so the exact classification and counterexample are the new contribution.
AI involvement
VibeMathed records the submitter’s account that GPT-6 Astra found the Bailey–Crandall question, produced the counterexample and exact criterion, and wrote the Lean development. The unbylined paper names no human author and does not independently verify that account.
VibeMathed records the submitter’s account that GPT-6 Astra found the Bailey–Crandall question, produced the counterexample and exact criterion, and wrote the Lean development. The unbylined paper names no human author and does not independently verify that account. The AI-led assessment rests on that submitter attribution; absence of a human byline alone does not prove autonomy.
Importance
Settles the Bailey–Crandall extension question and gives the exact normality criterion.
Limitations
The analytic sufficiency argument uses Bailey and Crandall’s prior estimate. The repository’s own dated prior-art check does not certify novelty. VibeMathed records a submitter attribution of the new classification and Lean work to AI; the unbylined repository does not independently identify the model as prover.
Technical verification
The pinned Lean 4 development reports `normal_iff` and `counterexample` with guarded standard-axiom audits and successful CI builds; VibeMathed reviewed the formal statements and original Bailey–Crandall question. This is a source-reported check of the exact selected theorem; AIProofWatch did not rerun the build.
Formal checking
The pinned Lean 4 development reports `normal_iff` and `counterexample` with guarded standard-axiom audits and successful CI builds; VibeMathed reviewed the formal statements and original Bailey–Crandall question.
- Checker
- Lean 4 / mathlib, source-reported CI; VibeMathed statement review
- Scope
- For integers b,c≥2, the Stoneham number α(b,c)=Σ_{k≥1}1/(c^k b^{c^k}) is normal in base b exactly when some prime divisor of c does not divide b. In particular α(6,4) is not 6-normal despite neither parameter dividing the other.
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Few is the default when evidence for a higher level is insufficient. The reviewed sources do not name a person who explains the central proof. Few is the default from missing documentation, not a finding that a person cannot understand it.
Correction and publication history
Primary paper, formal statement, prior work and submitter attribution reviewed for non-notifying local inclusion.
Read this published revision