Abstract
We study the (3x+1)/2 problem from a probabilistic viewpoint and show a forgetting mechanism for the lastk binary digits of the seed afterk iterations. The problem is subsequently generalized to a trifurcation process, the (lx+m)/3 problem. Finally the sequence of a set of seeds is empirically shown to be equivalent to a random walk of the variable log2 x (or log3 x) though computer simulations.
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References
G. T. Leavens and M. Vermeulen, 3x+1 search programs,Comput. Math. Appl. 24(11):79–99 (1992).
M. R. Feix, A. Muriel, D. Merlini, and R. Tartini, The (3x+1)/2 problem: A statistical approach, inProceeding of the 3rd International Conference: Stochastic Process in Physics and Geometry (Locarno, 1991).
D. A. Rawsthorne, Imitation of an iteration,Math. Mag. 58(3):172–176 (1985).
R. E. Crandall, On the 3x+1 problem,Math. Comp. 32(144):1281–1292 (1978).
H. Müller, Das 3x+1 problem,Mitt. Math. Ges. Hamburg 12(2):231–251 (1991).
J. C. Lagarias and A. Weiss, The 3x+1 problem: Two stochastic models,Ann. Appl. Prob. 2:229–261 (1992).
J. C. Lagarias, The 3x+1 problem and its generalization,Am. Math. Monthly 92:3–23 (1985).
R. Terras, A stopping time problem on the positive integers,Acta Arith. 30(3):241–256 (1976).
C. J. Everett, Iteration of the number theoretic functionf(2n)=n, f(2n+1)=3n+2, Adv. Math. 25(1):42–45 (1977).
J. C. Lagarias, The set of rational cycles for the 3x+1 problem,Acta Arith. 56(1):33–53 (1990).
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Feix, M.R., Muriel, A. & Rouet, J.L. Statistical properties of an iterated arithmetic mapping. J Stat Phys 76, 725–741 (1994). https://doi.org/10.1007/BF02188683
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DOI: https://doi.org/10.1007/BF02188683