Optimal bounds for the length of rational Collatz cycles

Halbeisen, Lorenz; Hungerbühler, Norbert

doi:10.4064/aa-78-3-227-239

Cited by 14 publications

(31 citation statements)

References 9 publications

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“…The word z k in v k D 1z k 0 is central and, by (Proposition 2.2.15, Lothaire [8]), the words s k WD z k 10 and s 0 k D z k 01 are standard words. 5 For k even, there are the bijections 5 For k even, there are the bijections…”

Section: Lemma 17 Limmentioning

confidence: 99%

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The 3x + 1 Conjugacy Map over a Sturmian Word

López¹,

Stoll²

2009

Integers

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The 3x C 1 map T is defined on the 2-adic integers Z 2 by T .x/ D x=2 for even x and T .x/ D .3x C 1/=2 for odd x. Under iteration of T , the sequence .T k .x/ mod 2/ 1 kD0 , called the parity vector of x 2 Z 2 , can be interpreted as an infinite word over the alphabet ¹0; 1º or as the digits of the 2-adic integerˆ 1 .x/ D P 1 kD0 .T k .x/ mod 2/ 2 k . For any v 2 Z 2 (or equivalently for the infinite word v), the inverse mapˆ(called the 3x C 1 conjugacy map) yields the unique x 2 Z 2 with parity vector v. It is unknown if there exists any aperiodic v with an eventually periodicˆ.v/. In this paper we computeˆ.v/ for a class of aperiodic infinite words v of minimal complexity, the mechanical words with irrational slope and intercept 0. Our main result is a generalized continued fraction expansion of 1=ˆ.x/, convergent under the 2-adic metric of Z 2 . The given examples suggest thatˆalways maps Sturmian words to infinite words of full complexity.

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Section: Lemma 17 Limmentioning

confidence: 99%

“…It is easy to prove that '.m p=q / is the same quantity as M`; n in[5], Corollary 1 (with`D q; n D p) 4. .0/ a1 1 means .a 1 1/ times 0.…”

mentioning

confidence: 91%

The 3x + 1 Conjugacy Map over a Sturmian Word

López¹,

Stoll²

2009

Integers

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“…Selbst wenn jede Collatzfolge für einen Startwert, der kleiner ist als 500 Billiarden, in diesen Zyklus mündet, ist bis heute nicht bewiesen, dass dies für alle möglichen natürlichen Startwerte zwangsläufig der Fall sein muss. Weiter ist bekannt, dass falls die Collatzfolge in einen Zyklus ohne die Zahl 1 münden würde, dieser Zyklus eine Länge hätte, dieüber 100 Millionen liegt [Halbeisen & Hungerbühler 1997]. Mit anderen Worten scheint diese mathematische Frage zwar eine plausible Antwort zu besitzen, sich jedoch bis heute einer endgültigen Beantwortung zu entziehen.…”

Section: Mögliche Mathematische Prozesse Während Der Phase Der Bespreunclassified

Mathematische Vorstellungen bilden — Praxis und Theorie von Vorstellungsübungen im Mathematikunterricht der Sekundarstufe II

Weber

2007

JMD

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“…Halbeisen and Hungerbühler [2] found new techniques which allow a refined analysis of rational (and hence integer) Collatz cycles. In particular, they prove optimal estimates for the length of a cycle having positive elements in terms of its minimum.…”

Section: Introductionmentioning

confidence: 99%

Generalizing Halbeisen's and Hungerbühler's optimal bounds for the length of Collatz cycles to 3 n + c cycles

Cox¹,

Ghosh²,

Sultanow³

2021

J. Math. Computer Sci.

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The next element in the 3n + 1 sequence is defined to be (3n + 1)/2 if n is odd or n/2 otherwise. The Collatz conjecture states that no matter what initial value of n is chosen, the sequence always reaches 1 (where it goes into the repeating sequence (1, 2, 1, 2, 1, 2, . . .)). The only known Collatz cycle is (1, 2). Let c be an odd integer not divisible by 3. Similar cycles exist for the more general 3n + c sequence. The 3n + c cycles are commonly grouped according to their length and number of odd elements. The smallest odd element in one of these cycles is greater than the smallest odd elements of the other cycles in the group. A parity vector corresponding to a cycle consists of 0's for the even elements and 1's for the odd elements. A parity vector generated by the ceiling function is used to determine this smallest odd element. Similarly, the largest odd element in one of these cycles is less than the largest odd elements of the other cycles in the group. A parity vector generated by the floor function is used to determine this largest odd element. This smallest odd element and largest odd element appear to be in the same cycle. This means that the parity vector generated by the floor function can be rotated to match the parity vector generated by the ceiling function. Two linear congruences are involved in this rotation. The natural numbers generated by one of these congruences appear to be uniformly distributed (after sorting).

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Optimal bounds for the length of rational Collatz cycles

Cited by 14 publications

References 9 publications

The 3x + 1 Conjugacy Map over a Sturmian Word

The 3x + 1 Conjugacy Map over a Sturmian Word

Mathematische Vorstellungen bilden — Praxis und Theorie von Vorstellungsübungen im Mathematikunterricht der Sekundarstufe II

Generalizing Halbeisen's and Hungerbühler's optimal bounds for the length of Collatz cycles to 3 n + c cycles

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