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.

**1.**A. Baker,*Transcendental Number Theory*, Cambridge University Press, Cambridge, 1975. MR**0422171 (54:10163)****2.**R.K. Guy,*Unsolved Problems in Number Theory*, 2nd ed., Springer Verlag, Berlin, 1994. MR**1299330 (96e:11002)****3.**G.H. Hardy and E.M. Wright,*The theory of Numbers*, 4th edition, Oxford University Press, Oxford, 1975.MR**0568909 (81i:10002)****4.**J.C. Lagarias,*The problem and its generalizations.*American Mathematical Monthly**92**(1985), 3-23.MR**0777565 (86i:11043)****5.**M. Laurent, M. Mignotte and Yu. Nesterenko,*Formes linéaires en deux logarithmes et déterminants d'interpolation.*Journal of Number Theory**55**(1995), 285-321. MR**1366574 (96h:11073)****6.**J.L. Simons and B.M.M. de Weger,*Theoretical and computational bounds for -cycles of the problem.*Accepted by Acta Arithmetica, 2004.**7.**R.P. Steiner,*A theorem on the Syracuse problem*, Proceedings of the 7th Manitoba Conference on Numerical Mathematics and Computation, 1977, pp. 553-559. MR**0535032 (80g:10003)****8.**M. Waldschmidt,*A lower bound for linear forms in logarithms.*Acta Arithmetica**37**(1980), 257-283. MR**0598881 (82h:10049)****9.**B.M.M. de Weger,*Algorithms for diophantine equations*, CWI Tract 65, Centre for Mathematics and Computer Science, Amsterdam, 1990.**10.**G.J. Wirsching,*The Dynamical System Generated by the Function*, Lecture Notes in Mathematics, Vol. 1681, Springer Verlag, Berlin, 1998. MR**1612686 (99g:11027)**

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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