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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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On the nonexistence of $2$-cycles for the $3x+1$ problem
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by John L. Simons PDF
Math. Comp. 74 (2005), 1565-1572 Request permission

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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Additional Information
  • John L. Simons
  • Affiliation: University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands
  • Email: j.l.simons@bdk.rug.nl
  • 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