Abstract

Collatz conjecture or 3N+1 problem by Lothar Collatz can be described as follows: For any positive integer N, a subsequent term is obtained from the previous term as follows: if N is even, the next term is one half of the previous term, but if the previous term is odd, then the next term is multiplied by 3 and added to 1. Repeat this process to the result over and over again. The conjecture is that no matter the positive integer N, the sequence will always reach 1. This paper attempts to present a new approach known as the “Unitary Series” approach, it is an approach that proves Collatz conjecture to be true. The key idea of this new approach focuses on the infinite loop (1, 4, 2) generated by any positive integer N if it indeed obeys Collatz algorithm and thus reducible to 1. This proof of Collatz algorithm for the problem is proposed and numerical example is given to show its feasibility. With this new approach, the proof of the conjecture is given.
Keyword: Collatz, 3n+1, conjecture, series, unitary.