An Asymptotic Approach for the Collatz Conjecture

Abstract

lt is well known that the Collatz Conjecture is one of the unsolved problems in mathematics. Supercomputer simulation has confirmed that the Collatz conjecture is correct for all positive integers unti1 2.95*10^20. Recently, Terence Tao proved that almost all orbits of the Collatz map attain almost bounded values. It is to say that the Collatz conjecture is almost correct for almost all positive integers. This paper proposes an asymptotic approach for the Collatz Conjecture by the mathematical induction and probability theory. The result of this paper means that the Collatz conjecture is correct for almost all positive integers. Keywords: Collatz Conjecture, asymptotic approach, probability, mathematical induction.