bibkey: chaffin2013curling authors: Benjamin Chaffin, John P. Linderman, N. J. A. Sloane, Allan R. Wilks year: 2013 title: “On curling numbers of integer sequences” doi: 10.48550/arXiv.1212.6102 url: https://arxiv.org/abs/1212.6102v3 claim: “The last sentence of Section 5 asks whether Theorem 23 survives when the starting sequence may contain a 1 but does not end with 1; Section 6 lists it as open question 12.” strata_touched:

• D5/S3/Combinatorics/ChaffinSloaneGijswijtMergeRefutation license: citation-only triage: anchor

Chaffin–Linderman–Sloane–Wilks curling numbers

Verified locator

DOI: 10.48550/arXiv.1212.6102

URL: https://arxiv.org/abs/1212.6102v3

Version: arXiv:1212.6102v3, the text that matches the journal article Journal of Integer Sequences 16 (2013), Article 13.4.3. Version 2 of the arXiv record numbers the conjectures differently and states a smaller verified range, so the version is load bearing. The authors and title match both the arXiv record and the journal record.

Definitions

Section 1 defines the curling number cn(S) of a finite integer sequence as the largest k such that S = X Y^k for some nonempty Y. For a starting sequence S_0 it sets S_{m+1} = S_m cn(S_m). When cn(S_t) = 1 holds for some t >= 0, the least such t is the tail length tau(S_0) and S^(e) := S_t is the extension of S_0. Section 5 writes G for Gijswijt’s sequence A090822, the continuation of the one-element sequence 1, and S^(inf) for the infinite continuation of S.

The statement in question

Section 5 states:

Theorem 23. Assume the curling number conjecture is true. Let S be an initial sequence not containing a 1, let S^(e) be its “extension” (defined in §1), and let S^(inf) be its infinite continuation. Then S^(inf) = S^(e) G.

The section closes with:

We do not know if the theorem is still true if S is allowed to contain a 1 but does not end with 1.

Section 6, which collects the article’s open questions, lists as item 12:

12. The question implicit in the last sentence of §5.

The prose immediately before Theorem 23 phrases the informal content as “any starting sequence S that does not contain a 1 must eventually merge with G”. That phrasing is weaker than the displayed equality; the question in item 12 refers to “the theorem”, whose conclusion is the displayed equality.

Where the published proof breaks

The proof of Theorem 23 writes S^(inf) = W(XT)^n n ⋯ with S^(e) = WX and T a prefix of G, and in the case n = 2 argues that the curling number of the first copy of T is the first term of X, “which is not 1”. That step uses the hypothesis that S^(e) contains no 1. Weakening the hypothesis to “contains a 1 but does not end with 1” keeps S^(e) from ending in 1 but does not keep X from beginning with 1.

Scope of the recorded answer

The starting sequence 1 2 contains a 1 and does not end with 1. Since cn(1 2) = 1, its tail length is 0 and its extension is 1 2 itself, so the extension exists without appeal to the curling number conjecture. Its continuation begins 1 2 1 1 2 1 2 2 2 3, whereas S^(e) G begins 1 2 1 1 2 1 1 2 2 2; the two differ at the seventh term because 1 2 1 1 2 1 = (1 2 1)^2 has curling number 2.

Because Theorem 23 is stated under the hypothesis that the curling number conjecture holds, the weakened statement read as an implication can still be true vacuously. The recorded result negates its consequent, so the weakened implication is equivalent to the negation of the curling number conjecture: a proof of the weakened Theorem 23 would refute that conjecture.

The weaker informal reading, whether S^(inf) has a suffix equal to a suffix of G, is a different statement and is not settled here. Sloane’s 2023 retrospective conjectures that merging holds for any starting sequence.

Bounded prior-resolution evidence

Issue 9218 records the screen carried out before the result was written up. An independent literature pass opened the journal article, the two-author Chaffin–Sloane preprint arXiv:0912.2382v5, Sloane’s 2023 retrospective, both arXiv versions and the journal version of Levi van de Pol’s growth-rate work, the OEIS entries A090822, A093369, A094004, A216955 and A217209 with the OEIS curling-number index, and several citing papers including Caveney, Dong and Shallit, arXiv:2608.15670v1. None of them records an answer to item 12. Major citation-index result pages were not reachable, so this is a bounded negative finding and no worldwide priority claim is made.