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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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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DOI: https://doi.org/10.1007/s11424-012-0280-5