The sufficiency of arithmetic progressions for the 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 by if is odd and if is even. The Conjecture states that the -orbit of every positive integer contains . A set of positive integers is said to be *sufficient *if the -orbit of every positive integer intersects the -orbit of an element of that set. Thus to prove the Conjecture it suffices to prove it on some sufficient set. Andaloro proved that the sets are sufficient for and asked if is also sufficient for larger values of . We answer this question in the affirmative by proving the stronger result that is sufficient for any nonnegative integers and with i.e. every nonconstant arithmetic sequence forms a sufficient set. We then prove analagous results for the Divergent Orbits Conjecture and Nontrivial Cycles Conjecture.

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

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Additional Information

**Kenneth M. Monks**

Affiliation:
Department of Mathematics, University of Scranton, Scranton, Pennsylvania 18510

Email:
monksk2@scranton.edu

DOI:
https://doi.org/10.1090/S0002-9939-06-08567-4

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