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.

- Paul Andaloro,
*On total stopping times under $3x+1$ iteration*, Fibonacci Quart.**38**(2000), no. 1, 73–78. MR**1738650** - Daniel J. Bernstein,
*A noniterative $2$-adic statement of the $3N+1$ conjecture*, Proc. Amer. Math. Soc.**121**(1994), no. 2, 405–408. MR**1186982**, DOI https://doi.org/10.1090/S0002-9939-1994-1186982-9 - Loo Keng Hua,
*Introduction to number theory*, Springer-Verlag, Berlin-New York, 1982. Translated from the Chinese by Peter Shiu. MR**665428** - Jeffrey C. Lagarias,
*The $3x+1$ problem and its generalizations*, Amer. Math. Monthly**92**(1985), no. 1, 3–23. MR**777565**, DOI https://doi.org/10.2307/2322189 - Michał Misiurewicz and Ana Rodrigues,
*Real $3x+1$*, Proc. Amer. Math. Soc.**133**(2005), no. 4, 1109–1118. MR**2117212**, DOI https://doi.org/10.1090/S0002-9939-04-07696-8 - Günther J. Wirsching,
*The dynamical system generated by the $3n+1$ function*, Lecture Notes in Mathematics, vol. 1681, Springer-Verlag, Berlin, 1998. MR**1612686**

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