Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-06-08567-4
Published electronically: May 9, 2006
MathSciNet review: 2231609
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $n\leq 4$ 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 $B\neq 0,$ i.e. every nonconstant arithmetic sequence forms a sufficient set. We then prove analagous results for the Divergent Orbits Conjecture and Nontrivial Cycles Conjecture.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11B25, 11B83

Retrieve articles in all journals with MSC (2000): 11B25, 11B83


Additional Information

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

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