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Statement

Setting: the notation of Lemma 5 and Theorem 4, with $n(l):=\lfloor l\log_32\rfloor$, and $p_k/q_k$ the convergents to $\log_23$ (Theorem 1, p. 9), of which the paper uses $p_{14}=301\,994$, $p_{16}=17\,087\,915$ and $p_{18}=102\,225\,496$ (p. 11).

Example (pp. 11--12, unnumbered). Suppose the Collatz conjecture is verified for all initial values $x_0\le m=212\,366\,032\,807\,211$. Then the length of a nontrivial Collatz cycle is at least $L=102\,225\,496$. The abstract (p. 1) and the introduction (p. 2) state the same result for Collatz cycles in $\mathbb N$ that do not contain $1$.

The paper compares $m$ with the verification bound $6.3\cdot10^{13}$ then reported (about $3.3$ times smaller, p. 2; "about three times", p. 11), and says that Eliahou's original criterion would need the value $2.9\cdot10^{14}$ for the same conclusion (pp. 2 and 12). The result is conditional on that verification, which the paper does not claim.

Source. Lorenz Halbeisen and Norbert Hungerbühler, Optimal bounds for the length of rational Collatz cycles, Acta Arith. 78 (1997), 227--239; the example on pp. 11--12 of the authors' preprint named on the source card, numbered 1--13 rather than by the journal's pagination.

Read depth. Claims checked: the statement and the four steps were read on the print. The computations they report (a Mathematica check and a direct evaluation of Corollary 1) are not printed and were not checked. Nothing here is independently reviewed.

Proof pointer

Pages 11--12, in four steps, aimed at the sufficient condition (5) (p. 3): $M_{l,n}/(2^l-3^n)\le m$ for all $n$ and $l<L$.

• First step: Theorem 2 or Theorem 3, with Eliahou's tables of $k$ or direct computation, gives $L\ge17\,087\,915$.
• Second step: a computer check of the bound of Proposition 1 (p. 6) against $m$ for every $l$ in $\{p_{16},\ldots,p_{18}-1\}$ outside $A_1=\{kp_{16}:k=1,\ldots,5\}$ and $A_2=\{kp_{16}+p_{14}:k=3,4,5\}$; the print cites "Lemma 1 or Remark 1" for this conclusion. The paper notes that the convergent $p_{17}$ causes no difficulty because $2^{p_{17}}-3^{q_{17}}$ is negative.
• Third step: a direct evaluation of Corollary 1 for $l=p_{16}$, $n=n(l)$; since $n(kl)=k\,n(l)$ for $k\le100$ at this length, the balanced sequence for $kl$ is $k$ copies of the one for $l$ and (4) gives the same quotient, which disposes of $A_1$. In this passage the print writes $q_{16}$ for the length called $p_{16}$ elsewhere. For $A_2$ the paper says only that it can be handled by a similar argument or by direct verification, without recording which was done.
• Fourth step: Lemma 7 (p. 9) extends the bound from $n=n(l)$ to all $n\le n(l)$ for every $l<p_{18}$.

Dependencies

Theorem 3 or Eliahou's Theorem 2 (p. 10), Proposition 1 (p. 6), Lemma 5 with Corollary 1 (p. 6), the decomposition formula (4) (p. 3) and Lemma 7 (p. 9).

Bears on

• #1135: background only. A cycle of the problem's $f$ in the positive integers other than $\{1,2\}$ would answer the problem in the negative; the example bounds the length of such a cycle from below, conditional on the stated verification, and does not exclude one.