Can the Collatz conjecture be proven, or not? Download the PDF version of this article. collatz.pdf 236 KB In 1937, shortly after the mathematician Lothar Collatz obtained his doctorate, he wrote down a problem in his notebook that later became known as Collatz’ problem or the \(3x+1)\-problem . The problem is remarkable since it is easy to state, but for more than eighty years no solution had been found.
The Collatz hypothesis is a theorem of the algorithmic theory of natural numbers. We prove the (algorithmic) formula that expresses the halting property of Collatz algorithm. The observation that Collatz's theorem cannot be proved in any elementary number theory completes the main result.
The 3x+1 problem concerns iteration of the map T(n) =(3n+1)/2 if n odd; n/2 if n even. The 3x +1 Conjecture asserts that for every positive integer n>1 the forward orbit of n includes the integer 1. This paper is an annotated bibliography of work done on the 3x+1 problem published from 2000 through 2009, plus some later papers that were preprints by 2009. This is a sequel to an annotated bibliography on the 3x+1 problem covering 1963-1999. At present the 3x+1 Conjecture remains unsolved.