The Dynamical System Generated by the 3n+1 Function

Wirsching, Günther

doi:10.1007/bfb0095985

Cited by 77 publications

(58 citation statements)

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“…We now present three technical lemmas that will be used in step two. We start by reproving a stronger version of [Wir,Lemma 3 …”

Section: Lemma 32 For Everymentioning

confidence: 99%

The sufficiency of arithmetic progressions for the 3𝑥+1 Conjecture

Monks

2006

Proc. Amer. Math. Soc.

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Abstract. Define T : Z + → Z + by T (x) = (3x + 1) /2 if x is odd and T (x) = x/2 if x is even. The 3x + 1 Conjecture states that the T -orbit of every positive integer contains 1. A set of positive integers is said to be sufficient if the T -orbit of every positive integer intersects the T -orbit of an element of that set. Thus to prove the 3x+1 Conjecture it suffices to prove it on some sufficient set. Andaloro proved that the sets 1 + 2 n N are sufficient for n ≤ 4 and asked if 1 + 2 n N is also sufficient for larger values of n. We answer this question in the affirmative by proving the stronger result that A + BN is sufficient for any nonnegative integers A and B with B = 0, i.e. every nonconstant arithmetic sequence forms a sufficient set. We then prove analagous results for the Divergent Orbits Conjecture and Nontrivial Cycles Conjecture.

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“…We now present three technical lemmas that will be used in step two. We start by reproving a stronger version of [Wir,Lemma 3 …”

Section: Lemma 32 For Everymentioning

confidence: 99%

The sufficiency of arithmetic progressions for the 3𝑥+1 Conjecture

Monks

2006

Proc. Amer. Math. Soc.

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“…It is interesting that this conjecture is equivalent to the positive answer to the Collatz "3n + 1 problem" [37], [38]. (Regarding 3n + 1 problem and, in particular, some its history consult [263] by J. G. Wirsching.)…”

Section: Miscellaneous Results and Problemsmentioning

confidence: 99%

Recent results on functional equations in a single variable, perspectives and open problems

Baron

2001

Aequationes Mathematicae

161

116

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“…Lagarias [3] has compiled a comprehensive annotated bibliography, which lists hundreds of publications related to this conjecture. A survey article and a monograph on Collatz' conjecture are [2] and [4], respectively.…”

Section: Introductionmentioning

confidence: 99%