New lower bounds for the size of a non-trivial loop in the Collatz 3x+1 and generalized px+q problem

TL;DR

This paper establishes new bounds on the size of non-trivial loops in the Collatz 3x+1 problem and its generalizations, providing insights into the structure and minimal elements of such loops.

Contribution

It introduces bounds on the minimal odd number in non-trivial Collatz loops and extends these bounds to the generalized px+q problem.

Findings

Bound on the minimal odd number in non-trivial loops

Calculation of the least number of odd elements for such loops

Generalized bounds for the px+q problem

Abstract

In the Collatz 3x+1 problem, there are 3 possibilities: Starting from any positive number, we either reach the trivial loop (1,4,2), end up in a non-trivial loop, or go until infinity. In this paper, we shall show that if a non-trivial loop with m odd numbers exists, then its minimum odd number is bounded above by a function of m. We shall also use that bound to calculate the least number of odd elements required for a non-trivial loop to exist. Also, the generalized bounds for the px+q problem are given.