# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## The sufficiency of arithmetic progressions for the $3x+1$ ConjectureHTML articles powered by AMS MathViewer

by Kenneth M. Monks
Proc. Amer. Math. Soc. 134 (2006), 2861-2872

## 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.
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