7x ± 1: Close Relative of the Collatz Problem
Brno University of Technology
Faculty of Information Technology
Brno, Czech RepublicE-mail: ibarina@fit.vutbr.cz
Received:
Received: 15 October 2022; revised: 11 December 2022; accepted: 13 December 2022; published online: 20 December 2022
DOI: 10.12921/cmst.2022.0000025
Abstract:
We show an iterated function of which iterates oscillate wildly and grow at a dizzying pace. We conjecture that the orbit of arbitrary positive integer always returns to 1, as in the case of the Collatz function. The conjecture is supported by a heuristic argument and computational results.
Key words:
References:
[1] J.C. Lagarias, The 3x + 1 problem: An annotated bibliography (1963– 1999), sorted by author, arXiv: math/0309224 (2003).
[2] J.C. Lagarias, The 3x + 1 problem: An annotated bibliography, II (2000-2009), arXiv: math/0608208 (2006).
[3] T. Tao, The Collatz conjecture, Littlewood-Offord theory, and powers of 2 and 3 (2011), https://terrytao.wordpress.com/2011/08/25/the-collatz-conjecture-littlewood-offord-theory-and-powers-of-2-and-3/.
[4] J.C. Lagarias, The 3x + 1 problem and its generalizations, The American Mathematical Monthly 92(1), 3–23 (1985).
[5] R.E. Crandall, On the “3x + 1” problem, Mathematics of Computation 32(144), 1281–1292 (1978).
[6] M.V.P. Garcia, F.A. Tal, A note on the generalized 3n + 1 problem, Acta Arithmetica 90(3), 245–250 (1999).
[7] F. Mignosi, On a generalization of the 3x + 1 problem, Journal of Number Theory 55(1), 28–45 (1995).
[8] K.R. Matthews, Generalized 3x + 1 mappings: Markov chains and ergodic theory, [In:] J.C. Lagarias (Ed.), The Ultimate Challenge: The 3x + 1 Problem, 79–103, AMS (2010).
We show an iterated function of which iterates oscillate wildly and grow at a dizzying pace. We conjecture that the orbit of arbitrary positive integer always returns to 1, as in the case of the Collatz function. The conjecture is supported by a heuristic argument and computational results.
Key words:
References:
[1] J.C. Lagarias, The 3x + 1 problem: An annotated bibliography (1963– 1999), sorted by author, arXiv: math/0309224 (2003).
[2] J.C. Lagarias, The 3x + 1 problem: An annotated bibliography, II (2000-2009), arXiv: math/0608208 (2006).
[3] T. Tao, The Collatz conjecture, Littlewood-Offord theory, and powers of 2 and 3 (2011), https://terrytao.wordpress.com/2011/08/25/the-collatz-conjecture-littlewood-offord-theory-and-powers-of-2-and-3/.
[4] J.C. Lagarias, The 3x + 1 problem and its generalizations, The American Mathematical Monthly 92(1), 3–23 (1985).
[5] R.E. Crandall, On the “3x + 1” problem, Mathematics of Computation 32(144), 1281–1292 (1978).
[6] M.V.P. Garcia, F.A. Tal, A note on the generalized 3n + 1 problem, Acta Arithmetica 90(3), 245–250 (1999).
[7] F. Mignosi, On a generalization of the 3x + 1 problem, Journal of Number Theory 55(1), 28–45 (1995).
[8] K.R. Matthews, Generalized 3x + 1 mappings: Markov chains and ergodic theory, [In:] J.C. Lagarias (Ed.), The Ultimate Challenge: The 3x + 1 Problem, 79–103, AMS (2010).
