Three Variations on a Theme by Collatz
نویسنده
چکیده
The Collatz problem is reformulated in three ways as a non-arithmetic problem in first-order logic.
منابع مشابه
Cellular Automata to More Efficiently Compute the Collatz Map
The Collatz, or 3x+ 1, Conjecture claims that for every positive integer n, there exists some k such that T (n) = 1, where T is the Collatz map. We present three cellular automata (CA) that transform the global problem of mimicking the Collatz map in bases 2, 3, and 4 into a local one of transforming the digits of iterates. The CAs streamline computation first by bypassing calculation of certai...
متن کاملThe Undecidability of the Generalized Collatz Problem
The Collatz problem, widely known as the 3x + 1 problem, asks whether or not a certain simple iterative process halts on all inputs. We build on earlier work by J. H. Conway, and show that a natural generalization of the Collatz problem is recursively undecidable.
متن کاملω-rewriting the Collatz problem
The Collatz problem is comprised of two distinct, separate questions: existence of eventually periodic Collatz reductions with a nontrivial period, and existence of period-free Collatz reductions. This paper introduces a few distinct, related formalizations of the Collatz dynamics as term rewriting systems, using elementary concepts from the theory of state transition dynamics. Some of the subj...
متن کاملThe Structure of the Collatz Graph; A Recursive Production of the Predecessor Tree; Proof of the Collatz 3x+1 Conjecture
Several major steps develop a proof of the Collatz conjecture. (1)The overall structure of the Collatz graph is a binary predecessor tree whose nodes contain residue sets, {c[d]}. (2)The residue sets are deployed as extensions (right descents) and l.d.a.s (left descents). (3)Generation of a large initial sample of the tree was accomplished using a recursive computer program. (4)The program outp...
متن کاملA New Proof of the Collatz Conjecture
This paper is a revised version of a previous paper by the author that, after publication, was found to contain an error. This paper supersedes and replaces the previous paper. A modified form of the Collatz transformation is formulated, leading to the concept of a modified Collatz chain. A smallest counterexample N0 to the Collatz Conjecture is hypothesized ; the existence of N0 implies that N...
متن کامل