Some contributions to Collatz conjecture

TL;DR

This paper explores new representations and functions related to the Collatz conjecture, aiming to characterize the set of integers that satisfy the conjecture and providing bounds on their binary representations.

Contribution

It introduces a new function ta(y) operating on rationals with finite binary expansion and shows binary representations of integers relate to powers of specific seed strings.

Findings

The function ta(y) simplifies analysis by limiting denominators to 2 or 4.

Binary representations of integers are prefixes of powers of infinitely many seed strings.

An upper bound for the seed index i in terms of binary length n is established.

Abstract

The Collatz conjecture can be stated in terms of the reduced Collatz function R(x) = (3x+1)/2^m (where 2^m is the larger power of 2 that divides 3x+1). The conjecture is: Starting from any odd positive integer and repeating R(x) we eventually get to 1. In a previous paper of the author the set of odd positive integers x such that R^k(x) = 1 has been characterized as the set of odd integers whose binary representation belongs to a set of strings G_k. Each string in G_k is the concatenation of k strings z_k z_{k-1} ... z_1 where each z_i is a finite and contiguous extract from some power of a string s_i of length 2x3^{i-1} (the seed of order i). Clearly Collatz conjecture will be true if the binary representation of any odd integer belongs to some G_k. Lately Patrick Chisan Hew showed that seeds s_i are the repetends of 1/3^i. Here two contributions to Collatz conjecture are given: - Collatz conjecture is expressed in terms of a function \rho(y) that operates on the set of all rational numbers 1/2 <= y < 1 having finite binary representation. The main advantage of \rho(y) with respect to R(x) is that the denominator can be only 2 or 4 (unlike R(x) whose denominator can be any power of 2). - We show that the binary representation of each odd positive integer x is a prefix of a power of infinitely many seeds s_i and we give an upper bound for the minimum i in terms of the length n of the binary representation of x.