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Statement

Setting (pp. 1--2): the Collatz operator $C:\mathbb Z_{>0}\to\mathbb Z_{>0}$, $C(n)=n/2$ for even $n$ and $(3n+1)/2$ for odd $n$ (Definition 1; the map $f$ of Problem 1135); an $m$-cycle is a nontrivial cycle of $C$ with $m$ local minima (Definition 5, p. 2: a cycle $n,C(n),\ldots,C^i(n)=n$ with $n>2$ and exactly $m$ local minima), so that its members fall into $m$ blocks, each a run of odd members followed by a run of even members; $K$ denotes the number of odd members of a cycle.

Theorem 23 (Main Theorem) (p. 15): "There is no $m$-cycle with $m\le91$."

The proof's first lines (p. 15): for $m\le91$, Theorem 3 of Simons and de Weger gives $K>7\cdot10^{11}$; with $X_0=704\cdot2^{60}\approx8.1\cdot10^{20}$ (Definition 4, the verification bound taken from Barina's project) one gets $m_2\ge47$, Theorem 21 gives $\delta<(K+L)/K<\delta+6.9\cdot10^{-32}$, and continued fractions (Lemma 22) give $K>5.2\cdot10^{15}$; iterating this process six more times ($m_2\ge67,77,82,86,88,91$) yields $K>7.94\cdot10^{21}$, and the proof closes on p. 16: "But this last lower bound on $K$ is larger than the upper bound of $K<1.4784\,m\delta^m<2.2\cdot10^{20}$ given by Simons and de Weger [12]. Thus, no such $m$-cycle can exist."

Source. C. Hercher, There are no Collatz m-cycles with $m\le91$, arXiv:2201.00406v3 (4 April 2023), the version retained; J. Integer Seq. 26 (2023), Article 23.3.5 (not compared). Theorem 23 and its proof on pp. 15--16 (PDF pp. 15--16), Definition 1 on p. 1, Definitions 4 and 5 and Remark 3 on p. 2, read on the rendered page images. The artifact is identified in the source digest.

Read depth. Claims checked: the statement and the proof's iteration were read clause by clause on the page image; the lemmas it invokes (Theorem 21, Lemma 22, the Simons--de Weger bounds) were read as statements in the text layer and not checked; the computations were not rerun.

Proof pointer

Sections 2--3 (pp. 3--16): with $L$ the number of even members, Theorem 16 bounds $(K+L)/K$ from above, and from below by $\delta=\log_23$, through the sums $T(n_i)$ of reciprocals of the run of odd members starting with each local minimum $n_i$; Theorem 21 sharpens the upper bound in terms of an integer $m_2\le m$ whose admissible size depends on $K$ and the verification bound $X_0$; Lemma 22 (a continued-fraction lemma: every fraction in an open interval has denominator at least that of a specified convergent) turns the bound into a lower bound for $K$; alternating the two steps raises the lower bound until it exceeds the Simons--de Weger upper bound $K<1.4784\,m\delta^m$. Not reconstructed here.

Dependencies

Simons and de Weger, Theoretical and computational bounds for $m$-cycles of the $3n+1$ problem, version 1.44 (2010), Theorem 3 (the lower bound $K>7\cdot10^{11}$) and the upper bound $K<1.4784\,m\delta^m$ (the paper's [12], a 2010 preprint; its [11] is the published Acta Arith. 117 (2005), 51--70, for which the paper records $m\ge68$, p. 2; card simons_de_weger_2010_mcycles_bounds, no file held, not consumed here); the verification bound $X_0=704\cdot2^{60}$ from Barina's project (the paper's [2]; the held 2020 paper reports $2^{68}$ and the project's later bound is the one used); continued fractions (Lemma 22, a well-known lemma proved in the paper).

Bears on

• Problem 1135: for the page's map $f$, no nontrivial cycle with at most $91$ local minima exists; the current cycle-exclusion record recorded on the page, which leaves cycles with more local minima and divergent trajectories open.