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

2021/05/13 by Fabian S. Reid, Reid, Fabian S.
Computer Science · Mathematics · #Benford’s Law and Fraud Detection #Computability, Logic, AI Algorithms #math.GM

paper · pdf · doi:10.48550/arxiv.2105.07955

22 pages

arxiv created 2021/05/13 · arxiv updated 2021/05/18

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.

Related

• Encoding and Visualization in the Collatz Conjecture
• On the nonexistence of cycles for the Collatz function
• The Collatz Network
• Observations on cycles in a variant of the Collatz Graph
• An Exploration Into the Collatz Conjecture with Changed Parameters