On the nonexistence of cycles for the problem
Author:
John L. Simons
Journal:
Math. Comp. 74 (2005), 15651572
MSC (2000):
Primary 11J86, 11K60; Secondary 11K31
Published electronically:
December 8, 2004
MathSciNet review:
2137019
Fulltext PDF Free Access
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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:
http://dx.doi.org/10.1090/S0025571804017284
PII:
S 00255718(04)017284
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
