Some contributions to Collatz conjecture

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.