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On the nonexistence of nontrivial small cycles of the µ function in 3x+1 conjecture

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Abstract

This paper studies the property of the recursive sequences in the 3x + 1 conjecture. The authors introduce the concept of µ function, with which the 3x + 1 conjecture can be transformed into two other conjectures: one is eventually periodic conjecture of the µ function and the other is periodic point conjecture. The authors prove that the 3x + 1 conjecture is equivalent to the two conjectures above. In 2007, J. L. Simons proved the non-existence of nontrivial 2-cycle for the T function. In this paper, the authors prove that the µ function has no l-periodic points for 2 ≤ l ≤ 12. In 2005, J. L. Simons and B. M. M de Weger proved that there is no nontrivial l-cycle for the T function for l ≤ 68, and in this paper, the authors prove that there is no nontrivial l-cycle for the µ function for 2 ≤ l ≤ 102.

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Author information

Correspondence to Dengguo Feng.

Additional information

This research is supported by Natural Science Foundation of China under Grant Nos. 60833008 and 60902024.

This paper was recommended for publication by Editor Lei HU.

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Feng, D., Fan, X., Ding, L. et al. On the nonexistence of nontrivial small cycles of the µ function in 3x+1 conjecture. J Syst Sci Complex 25, 1215–1222 (2012). https://doi.org/10.1007/s11424-012-0280-5

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Key words

  • Diophantine equation
  • eventual period
  • periodic point
  • 3x +1 conjecture