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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

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