Generating the Convergence Stairs of the Collatz Program

Document Type

Conference Proceeding

Publication Date

2024

Department

Department of Computer Science

Abstract

For the first time in decades, this paper presents an algorithmic method that, given a positive integer j, generates the j-th convergence stair containing all natural numbers from where the Collatz conjecture holds by exactly j applications of the Collatz function. To this end, we present a novel formulation of the Collatz conjecture as a concurrent program, and provide the general case specification of the j-th convergence stair for any j>0. The proposed specifications provide a layered and linearized orientation of Collatz numbers organized in an infinite set of infinite binary trees. Such a general specification can have significant applications in analyzing and testing the stability of complex non-linear systems. We have implemented this method as a software tool that generates the Collatz numbers of individual stairs. We also show that starting from any value in any convergence stair the conjecture holds. However, to prove the conjecture, one has to show that every natural number will appear in some stair; i.e., the union of all stairs is equal to the set of natural numbers, which remains an open problem.

Publication Title

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

ISBN

9783031744976

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