Statistical Analysis of Descending Open Cycles of Collatz Function

Kamal Barghout (), Wadii Hajji, Nidal Abu-Libdeh and Mohammad Al-Jamal
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Kamal Barghout: Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, Al Khobar 31952, Saudi Arabia
Wadii Hajji: Department of Mathematics, University of Ottawa, Ottawa, ON K1N 6N5, Canada
Nidal Abu-Libdeh: Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, Al Khobar 31952, Saudi Arabia
Mohammad Al-Jamal: Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, Al Khobar 31952, Saudi Arabia

Mathematics, 2023, vol. 11, issue 3, 1-15

Abstract: Collatz dynamic systems present a statistical space that can be studied rigorously. In a previous study, the author presented Collatz space in a unique dynamic numerical mode by tabulating a sequential correlation pattern of division by 2 of Collatz function’s even numbers until the numbers became odd with a consecutive occurrence, following an attribute of a 50:50 probability of division by 2 once (ascending behavior) as opposed to division by 2 more than once (descending behavior). In this paper, we describe the path of the Collatz function as sequences comprised of groups of the function’s iterates (open cycles) that end up with the first odd integer that is less than the starting odd integer. The descending behavior of the open cycles is attributed to a deterministic factor as observation of the cycles’ sequences shows. We do statistical analysis on 4 large samples of open cycles and orbits to 1. We define R ( n ) as the cycles’ deterministic variable defined as the ratio of division by 2 once to division by 2 more than once. We use statistical analysis to study the randomness of the orbits of the cycles’ starting odd positive integers as well as orbits to 1 up to 1,002,097,149.

Keywords: Collatz conjecture; statistical analysis; probabilistic model (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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