Knight 2026 high cycles

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Kevin Knight, Collatz High Cycles Do Not Exist, HAL hal-04261183 (preprint dated 23 September 2023, submitted to HAL 26 October 2023; 10 pp. with HAL cover, printed pages 1--9 on PDF pp. 2--10); published as Collatz high cycles do not exist, Discrete Math. 349 (2026), no. 3, Article 114812, DOI 10.1016/j.disc.2025.114812 (the Crossref record; the journal text is not held and was not compared).

Works with the shortcut map T. The cycles are taken over the rationals (a fraction counts as odd by its numerator); among the cycles of a given length $k$ with a given number $x$ of odd terms, the circuit is the one containing the least of their smallest members and the high cycle the one containing the greatest (abstract and p. 2). Steiner (1977) showed that no circuit of positive integers exists except 1-2-1 (pp. 2--3; reproved as Theorem 3.4); Theorem 5.4 (Main result, p. 8) states that no high cycle consists of integers, by exhibiting two members whose combination $3f(\mathbf v_h)-f(\mathbf v_h^R)+1$ equals $2^{k-1}/(2^k-3^x)$. The final step treats $2^k-3^x$ as an odd number greater than $1$; for one odd term ($k=2$, $x=1$, $2^k-3^x=1$) the high cycle is the trivial cycle 1-2-1, an exception the theorem's wording does not state. Statements read on the page images of pp. 1--2 and 8, the rest of the Steiner sentence (p. 3) in the text layer; proofs not checked. Relevance: Excludes Collatz high cycles, a class of potential counterexample cycles for problem 1135.

Source: PDF. The file prints "Distributed under a Creative Commons CC BY 4.0 - Attribution - International License" on its HAL cover page (PDF p. 1), with "HAL Id: hal-04261183, https://hal.science/hal-04261183v1": the Creative Commons Attribution 4.0 license.

Bears on. #1135