The Visual Pattern in the Collatz Conjecture and Proof of No Non-Trivial Cycles

TL;DR

This paper uncovers a visual pattern in the Collatz conjecture using a logarithmic spiral, linking it to primes and proving the absence of non-trivial cycles, advancing understanding of this longstanding problem.

Contribution

It introduces a novel visual pattern in the Collatz problem and proves that no non-trivial cycles exist, connecting the problem to prime numbers through Jacobsthal numbers.

Findings

Identified a logarithmic spiral pattern in the Collatz problem

Linked the Collatz problem to primes via Jacobsthal numbers

Proved the non-existence of non-trivial cycles in the Collatz conjecture

Abstract

We present the long sought visual pattern in the Collatz problem with the aid of a logarithmic spiral. Using this newly discovered pattern, we show that the Collatz problem is linked to primes via Jacobsthal numbers. We then prove that no non-trivial cycles exist on the spiral and by extension in the Collatz problem.