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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

   

 

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


Author: Kenneth M. Monks
Journal: Proc. Amer. Math. Soc. 134 (2006), 2861-2872
MSC (2000): Primary 11B25, 11B83
Published electronically: May 9, 2006
MathSciNet review: 2231609
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Abstract: Define $ T:\mathbb{Z} ^{+}\rightarrow\mathbb{Z} ^{+}$ by $ T\left( x\right) =\left( 3x+1\right) /2$ if $ x$ is odd and $ T\left( x\right) =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} \mathbb{N}$ are sufficient for $ n\leq4$ and asked if $ 1+2^{n}\mathbb{N}$ is also sufficient for larger values of $ n$. We answer this question in the affirmative by proving the stronger result that $ A+B\mathbb{N} $ is sufficient for any nonnegative integers $ A$ and $ B$ with $ B\neq0,$ 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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Additional Information

Kenneth M. Monks
Affiliation: Department of Mathematics, University of Scranton, Scranton, Pennsylvania 18510
Email: monksk2@scranton.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08567-4
PII: S 0002-9939(06)08567-4
Keywords: $3x+1$ problem, arithmetic sequences, orbits
Received by editor(s): May 1, 2005
Published electronically: May 9, 2006
Communicated by: Michael Handel
Article copyright: © Copyright 2006 Kenneth M. Monks