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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L. Collatz, On the origin of the 3x+1 problem, Natural Science Edition, 1986, 12(3): 9–11.
J. C. Lagarias, The 3x + 1 problem and its generalizations, Amer. Math. Monthly, 1985, 92: 3–23.
R. K. Guy, Unsolved Problems in Number Theory (Second Edition), Springer, Berlin, 1944.
G. J. Wirsching, The Dynamical System Generated by the 3n + 1 Function, Springer-Verlag, New York, 1998.
K. R. Matthews, The generalized 3x+1 Mapping, http://www.Numbertheory.org/pdfs/survey.pdf.
R. N. Buttsworth and K. R. Matthews, On some Markov matrices arising from the generalized Collatz mapping, Acta Arith., 1900, 55: 43–57.
G. M. Leigh, A Markov process underlying the generalized Syracuse algorithm, Acta Arith., 1985, 46: 125–143.
K. R. Matthews, Some Borel measures associated with the generalized Collatz mapping, Colloquium Math., 1992, 63: 191–202.
K. R. Matthews and G. M. Leigh, A generalization of the Syracuse algorithm in Fq[x], J. Number Theory, 1987, 25: 274–278.
K. R. Matthews and A. M. Watts, A generalization of Hasses’s generalization of the Syracuse algorithm, Acta Arith., 1983, 43: 167–175.
K. R. Matthews and A. M. Watts, A Markov approach to the generalized Syracuse algorithm, Acta Arith., 1985, 45: 29–42.
J. L. Simons and B. M. M. de Weger, Theoretical and computational bounds for m-cycles of the 3n + 1 problem, Acta Arith., 2005, 117: 51–70.
John L. Simons, On the non-existence of 2-cycles for the 3x +1 problem, Math. Comp., 2005, 74: 1565–1572.
John L. Simons, A simple (inductive) proof for the non-existence of 2-cycles of the 3x+1 problem, Journal of Number Theory, 2007, 123: 10–17.
T. Oliveirae Silva, Computational Verification of the 3x+1 conjecture. http://www.ieeta.pt/~tos/3x+1.html
C. Bohm and G. Sontacchi, On the existence of cycles of given length in integer sequences like x n + 1 = x n/2, if x n even, and x n + 1 = 3x n + 1, Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Natur., 1978, 64: 260–264.
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
- Diophantine equation
- eventual period
- periodic point
- 3x +1 conjecture