Abstract
The Collatz dynamic is known to generate a complex quiver of sequences over natural numbers for which the inflation propensity remains so unpredictable it could be used to generate reliable proof-of-work algorithms for the cryptocurrency industry; it has so far resisted every attempt at linearizing its behavior. Here, we establish an ad hoc equivalent of modular arithmetics for Collatz sequences based on five arithmetic rules that we prove apply to the entire Collatz dynamical system and for which the iterations exactly define the full basin of attractions leading to any odd number. We further simulate these rules to gain insight into their quiver geometry and computational properties and observe that they linearize the proof of convergence of the full rows of the binary tree over odd numbers in their natural order, a result which, along with the full description of the basin of any odd number, has never been achieved before. We then provide two theoretical programs to explain why the five rules linearize Collatz convergence, one specifically dependent upon the Axiom of Choice and one on Peano arithmetic.
Highlights
Accepted: 29 July 2021In 1937, Lothar Collatz established a conjecture known as the 3n + 1 problem, known as Kakutani’s problem, the Syracuse algorithm, Hasse’s algorithm, Thwaites conjecture, and Ulam’s problem
Our most essential theorems consist of the five rules that exactly define the basin of attraction of any odd number in the Collatz dynamical system
This is what we attempted in this article, primarily by establishing an ad hoc equivalent of modular arithmetic for Collatz sequences to automatically demonstrate the convergence of infinite quivers of numbers based on five arithmetic rules we proved by application in the entire dynamical system and which we further simulated to gain insight into their graph geometry and computational properties
Summary
In 1937, Lothar Collatz established a conjecture known as the 3n + 1 problem, known as Kakutani’s problem, the Syracuse algorithm, Hasse’s algorithm, Thwaites conjecture, and Ulam’s problem. Our methodology consists of using the complete binary tree and the complete ternary tree (the complete binary tree over odd numbers is defined as 2N∗ + 1 endowed with the following two linear applications {·2 − 1; ·2 + 1} and all their possible combinations, with the complete ternary tree over the same set in turn defined as 2N∗ + 1 endowed with operations {·3 − 2; ·3; ·3 + 2} and all their possible combinations) over 2N∗ + 1 as a general coordinate system for each node of the feather We owe this strategy to earlier discussions with Feferman [25] on his investigations on the continuum hypothesis, as it is known that the complete binary tree over natural numbers is one way of generating real numbers. Central to our contribution to the Collatz conjecture in this paper is the analysis of the branching factor of a zero-player cellular game developing in the complete binary tree over odd numbers
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