Available in electronic format
Available in print format		 Mathematics of Computation
Journal of the American Mathematical Society
ISSN 1088-6842(e) ISSN 0025-5718(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

On the nonexistence of $2$-cycles for the $3x+1$ problem

Author(s): John L. Simons.
Journal: Math. Comp. 74 (2005), 1565-1572.
MSC (2000): Primary 11J86, 11K60; Secondary 11K31
Posted: December 8, 2004
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

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.

References:

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 $3x+1$ 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 $m$-cycles of the $3n+1$ 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 $3n+1$ Function, Lecture Notes in Mathematics, Vol. 1681, Springer Verlag, Berlin, 1998. MR 1612686 (99g:11027)

Similar Articles:

Retrieve articles in Mathematics of Computation with MSC (2000): 11J86, 11K60, 11K31

Retrieve articles in all Journals with MSC (2000): 11J86, 11K60, 11K31

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: 10.1090/S0025-5718-04-01728-4
PII: S 0025-5718(04)01728-4
Keywords: 3x+1 problem, cycles, linear form in logarithms, continued fractions
Received by editor(s): February 13, 2003 and in revised form, May 4, 2004
Posted: December 8, 2004
Copyright of article: Copyright 2004, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google