publication . Article . 2021

Two-Parameter Generalization of the Collatz Function: Characterization of Terminal Cycles and Empirical Results

Boris B. Stefanov;
Open Access English
  • Published: 17 Mar 2021
  • Publisher: Zenodo
This paper proposes a new two-parameter generalization T_{b,s}(x) of the Z -> Z Collatz function T(x) and restates the eponymous conjecture in terms of the proposed function. The generalization obviates some of the conditions for emergence of terminal cycles for the Collatz T(x) function over the integers. The stopping behavior of the T_{b,s}(x) is qualitatively similar to that of the T(x). The paper presents theoretical discussion of the generalization and computational results on the terminal cycles and stopping times of T_{b,s}(x). The {1,2} cycle of T(x) is shown to be a case of coincidence of three independent cycle categories of T_{b,s}(x).
free text keywords: Collatz conjecture, 3x+1 problem, iteration, convergence, terminal cycles
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Open Access
Article . 2021
Providers: ZENODO
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