There are no cycles in the $3n+1$ sequence

TL;DR

This paper proves that the Collatz sequence has no cycles other than the known trivial cycle using graph theory and linear algebra, supporting the conjecture that all sequences eventually reach 1.

Contribution

The paper introduces a novel proof that there are no non-trivial cycles in the Collatz sequence employing graph theoretical and linear algebraic methods.

Findings

No cycles other than the trivial one are possible in the Collatz sequence.

The proof uses graph theory to model the sequence transitions.

Linear algebra techniques confirm the absence of non-trivial cycles.

Abstract

In 1937, Lothar Collatz conjectured that the sequence generated by the rule $f(n)=3n+1$ for $n\in\mathbb{N}$ odd, $f(n)=n/2$ for $n\in\mathbb{N}$ even, starting in any positive integer $n$ produces $1$. This is equivalent to (1) there are no cycles except the trivial one, (1-4-2-1), and (2) there is no infinite sequence. We prove (1) using graph theory and linear algebra.