The 3x+1 problem: new lower bounds on nontrivial cycle lengths

Eliahou, Shalom

doi:10.1016/0012-365x(93)90052-u

Cited by 29 publications

(46 citation statements)

References 3 publications

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“…Proof. This result is straightforward to prove by applying the same method as in [4]. Indeed, we have…”

Section: Hypothesis 23 (Lower Bound Hypothesis -Lbh)mentioning

confidence: 64%

The $3x+1$ problem: a lower bound hypothesis

Rozier¹

2017

Funct. Approx. Comment. Math.

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Much work has been done attempting to understand the dynamic behaviour of the so-called "3x + 1" function. It is known that finite sequences of iterations with a given length and a given number of odd terms have some combinatorial properties modulo powers of two. In this paper, we formulate a new hypothesis asserting that the first terms of those sequences have a lower bound which depends on the binary entropy of the "ones-ratio". It is in agreement with all computations so far. Furthermore it implies accurate upper bounds for the total stopping time and the maximum excursion of an integer. Theses results are consistent with two previous stochastic models of the 3x + 1 problem.Conjecture 1.1. (3x+1 problem) For any integer n > 0, we have T (j) (n) = 1 for some j ≥ 0, where T (j) denotes the j-th iterate of T .The 3x + 1 problem may be divided into Conjectures 1.2 and 1.3 below, asserting the absence of any other dynamic than the trivial cycle. Conjecture 1.2. (Absence of divergent trajectory) For all positive integer n, the infinite sequence T (k) (n) ∞ k=0 , called the trajectory of n, is bounded. Conjecture 1.3. (Absence of non-trivial cycle) There exist no integers n > 2 and j > 0 such that T (j) (n) = n.

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“…Proof. This result is straightforward to prove by applying the same method as in [4]. Indeed, we have…”

Section: Hypothesis 23 (Lower Bound Hypothesis -Lbh)mentioning

confidence: 64%

The $3x+1$ problem: a lower bound hypothesis

Rozier¹

2017

Funct. Approx. Comment. Math.

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“…Lemma 1, Assumptions (1.3.1), and Theorem 1, along with the bounds provided in [13], [5], and [7], demonstrate the non-existence of circuits in the 3x + 1 dynamical system.…”

Section: Circuits With the 3x + 1 Dynamical Systemmentioning

confidence: 79%

A Dual-Radix Approach to Steiner’s 1-Cycle Theorem

Rukhin

2019

Combinatorial and Additive Number Theory III

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“…We want to prove that then the length of a non-trivial Collatz cycle is at least L = 102 225 496. F i r s t s t e p: Using Theorem 2 or 3 and either the tables in [3] which list the function k or direct computation, we conclude that L ≥ 17 087 915. Observe that q 16 log 3 2 = q + ε for some number q ∈ N and ε ∈ ]0, 1/100[ and this implies that kn(q 16 ) = n(kq 16 ) for k = 1, 2, .…”

Section: Optimal Criterionmentioning

confidence: 85%

“…In [3] Eliahou improved Crandall's estimate and obtained the following criterion. Using the theory of continued fractions Eliahou showed that k(2 40 ) = 17 087 915 (this can also be checked directly from the definition).…”

Section: Eliahou's Criterionmentioning

confidence: 99%