Number Theory

    

Proof of the Collatz Conjecture Based on a Directed Graph

Authors: Wiroj Homsup, Nathawut Homsup

The Collatz conjecture considers recursively sequences of positive integers where n is succeeded by n/2 , if n is even or (3n+1)/2 , if n is odd. The conjecture states that for all starting values n the Collatz sequence eventually reaches a trivial cycle 1, 2, 1, 2u2026u2026. If the Collatz conjecture is false, then either there is a nontrivial cycle, or one sequence goes to infinity. In this paper, we construct a directed graph based on the union of infinite number of basic Collatz directed graphs. Each basic Collatz directed graph relates to each positive integer. We show that the directed graph is connected and covers all positive integers. There is only a trivial cycle and no sequence goes to infinity.

Comments: 6 Pages. (Note by viXra Admin: For the last time, both author's names - Not in call caps -should be filled on the Replacement Form)

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Submission history

[v1] 2024-09-07 03:27:25
[v2] 2024-09-12 20:46:33
[v3] 2024-10-18 20:04:01

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