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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

On the nonexistence of $2$-cycles for the $3x+1$ problem


Author: John L. Simons
Journal: Math. Comp. 74 (2005), 1565-1572
MSC (2000): Primary 11J86, 11K60; Secondary 11K31
Published electronically: December 8, 2004
MathSciNet review: 2137019
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Abstract: This article generalizes a proof of Steiner for the nonexistence of $1$-cycles for the $3x+1$ problem to a proof for the nonexistence of $2$-cycles. A lower bound for the cycle length is derived by approximating the ratio between numbers in a cycle. An upper bound is found by applying a result of Laurent, Mignotte, and Nesterenko on linear forms in logarithms. Finally numerical calculation of convergents of $\log_2 3$ shows that $2$-cycles cannot exist.


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Additional Information

John L. Simons
Affiliation: University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands
Email: j.l.simons@bdk.rug.nl

DOI: http://dx.doi.org/10.1090/S0025-5718-04-01728-4
PII: S 0025-5718(04)01728-4
Keywords: 3x+1 problem, cycles, linear form in logarithms, continued fractions
Received by editor(s): February 13, 2003
Received by editor(s) in revised form: May 4, 2004
Published electronically: December 8, 2004
Article copyright: © Copyright 2004 American Mathematical Society