Abstract
The unsolved number theory problem known as the 3x + 1 problem involves sequences of positive integers generated more or less at random that seem to always converge to 1. Here the connection between the first integer (n) and the last (m) of a 3x + 1 sequence is analyzed by means of characteristic zero-one strings. This method is used to achieve some progress on the 3x + 1 problem. In particular, the long-standing conjecture that nontrivial cycles do not exist is virtually proved using probability theory.
Highlights
Everett [1] (Iteration of the number-theoretic function f(2n) = n, f(2n + 1) = 3n + 2) introduced the concept of parity vectors to obtain early results concerning the 3x + 1 problem
One must prove that every Collatz sequence Ck(n) of finite length k converges to 1 for all positive integers n, k large enough
It is clear that each positive integer is a generator of a string, but not so clear that a given string will have a generator
Summary
This generates the Collatz sequence {n = n1, n2 , } (named after Lothar Collatz who introduced the problem). One must prove that every Collatz sequence Ck(n) of finite length k converges to 1 for all positive integers n, k large enough. This has been numerically verified for all n < n∗ = 20 × 258 = 5.7646 ×1018 (Lagarias [2]), and more recently to n=∗ 8.7 ×1018. At first we consider only finite strings (corresponding to finite Collatz sequences) At this point, it is clear that each positive integer is a generator of a string, but not so clear that a given (finite) string will have a generator.
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