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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- John L. Simons
- Affiliation: University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands
- Email: firstname.lastname@example.org
- Received by editor(s): February 13, 2003
- Received by editor(s) in revised form: May 4, 2004
- Published electronically: December 8, 2004
- © Copyright 2004 American Mathematical Society
- Journal: Math. Comp. 74 (2005), 1565-1572
- MSC (2000): Primary 11J86, 11K60; Secondary 11K31
- DOI: https://doi.org/10.1090/S0025-5718-04-01728-4
- MathSciNet review: 2137019