Halbeisen hungerbuhler 1997 optimal rational cycle bounds

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example_p11: The paper's worked example of Theorem 4: if the Collatz conjecture holds for all initial values up to 212366032807211, then every Collatz cycle in the positive integers not containing 1 has length at least 102225496.

lemma_5: For natural numbers n at most l, the 0-1 sequence whose first k terms sum to the ceiling of kn/l attains the largest possible minimum of phi over a rotation class in S_{l,n}, and Corollary 1 (p. 6) writes that maximum M_{l,n} as an explicit sum.

lemma_9: For a nonzero 0-1 sequence s with 2^l - 3^n positive, the greatest common divisor of phi over the rotations of s equals the gcd of phi(s) and 2^l - 3^n, from which the paper restates the existence of a nontrivial integer Collatz cycle as a divisibility condition (A').

theorem_3: Every positive Collatz cycle C over the rationals with odd denominator with more than 160 elements has length at least k(min C / alpha), where alpha is 0.9 and k(m) is the least k with k/n(k) at most log_2(3 + 1/m).

theorem_4: Every positive Collatz cycle C over the rationals with odd denominator has length at least L(min C), where L(m) is the least L for which M_{L,n(L)} divided by 2^L - 3^{n(L)} is at least m.

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Lorenz Halbeisen and Norbert Hungerbühler, Optimal bounds for the length of rational Collatz cycles, Acta Arith. 78 (1997) 227-239, https://doi.org/10.4064/aa-78-3-227-239 (author copy, people.math.ethz.ch/~halorenz/publications; 13 pp.).

Studies Collatz cycles over the rationals with odd denominator and bounds the length of a positive cycle from below by a function of its minimum element, with bounds attained for every length (Theorem 4 and Lemma 5); a sharpened form of Eliahou's criterion (Theorem 3, for positive cycles of length over 160) lets the conjecture be checked to a value about 10% smaller than Eliahou's for the same cycle-length bound. As an example, the length of a Collatz cycle in $\mathbb{N}$ not containing 1 is at least 102 225 496 provided the Collatz conjecture holds for all $x_0\le 212\,366\,032\,807\,211$, a range about 3.3 times the bound $6.3\cdot10^{13}$ then verified, so the bound was conditional when printed. Relevance: a nontrivial integer cycle would be a counterexample to problem 1135; the paper constrains such cycles (their length, given a verified range) and does not exclude them.

Source: PDF. The copy read for this card is the authors' preprint, which prints no copyright or license line on its first or last page and no journal header; no publisher's or repository's record was read for it; the term is unstated.

Bears on. #1135: the paper's maps $g_0(x)=x/2$ and $g_1(x)=(3x+1)/2$ are the problem's $f$ extended to the rationals with odd denominator, and a cycle of $f$ in the positive integers other than $\{1,2\}$ would answer the problem in the negative. Theorems 3 and 4 bound the length of such a cycle from below in terms of its minimum, the worked example does so conditionally on a verification the paper does not claim, and Lemma 9 yields the paper's reformulation (A$'$) of the existence of such a cycle as a divisibility condition; none of them settles the problem.

Results. Lemma 5 (p. 5), with Corollary 1 (p. 6), the sequence attaining $M_{l,n}$, the largest over $S_{l,n}$ of the least value of $\varphi$ on a rotation class; Theorem 3 (p. 10), Eliahou's criterion improved by the factor $0.9$; Theorem 4 (p. 11), the optimal criterion $|C|\ge L(\min C)$; the example (pp. 11--12), the conditional bound $102\,225\,496$; Lemma 9 (p. 12), the gcd identity behind the reformulation (A$'$) (p. 13). Pages are those of the preprint read, numbered 1--13.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.