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Volume 28 (4) 2022, 143–147

7x ± 1: Close Relative of the Collatz Problem

Barina David

Brno University of Technology
Faculty of Information Technology
Brno, Czech Republic

E-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:

Collatz problem, number theory

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).

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