On the nonexistence of -cycles for the problem

Author:
John L. Simons

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

Published electronically:
December 8, 2004

MathSciNet review:
2137019

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Abstract | References | Similar Articles | Additional Information

Abstract: This article generalizes a proof of Steiner for the nonexistence of -cycles for the problem to a proof for the nonexistence of -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 shows that -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:
https://doi.org/10.1090/S0025-5718-04-01728-4

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