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

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.