Trial for a proof of the Syracuse conjecture

TL;DR

This paper attempts to prove the Syracuse (Collatz) conjecture by disproving its two main antitheses, using algebra and graph theory, aiming to advance understanding of this longstanding mathematical problem.

Contribution

It introduces a novel approach to the Collatz conjecture by systematically disproving its key counterexamples through algebraic and graph-theoretic methods.

Findings

Disproved the existence of an unbounded Syracuse sequence

Disproved the existence of a cycle other than 4,2,1

Provided new insights into the structure of the conjecture

Abstract

The infamous 3x+1 conjecture spread by Lothar Collatz in 1952, despite its elementary formulation, remained unproved for over 60 years. From the heuristical probabilistic approach to the complex mapping of the algorithm, the scientific community has fetched for many methods to try to prove it formally, and thus, mathematicians like Erdos tend to believe that "mathematics are not yet ready for such problems". In this research report, covering domains like algebra and graph theory, it is shown a trial of proof of the conjecture by disproval of its two antitheses: the existence of an ever-growing Syracuse suite and the existence of a cycle different from the cycle 4,2,1.