Proof that Collatz conjecture is positive using the classiification in binary, the general term of progression of differences and graph theory

Abstract

This paper presents a new proof of Collatz conjecture using the classification in binary, a multiple of 3, the general term of progression of differences and graph theory. When Collatz process is done, we focus on numbers. Many sequences of numbers are generated. It is the progression of differences. The general term of progression of differences are computed. Then, the proof of contradiction, a multiple of 3, the general term of progression of differences and the Inverse-Collatz process are used to prove that all positive odd other than one do not enter an infinite loop (e.g. 1 → 3 → 4 → 1) by Collatz process. Using the classification in binary, we focus on the number of digits. We calculate the expected values of digit (multiply 3 and add 1) (A) and (divide by 2) (B). Comparing the expected values of A and B, we find that there are unequal (B is greater than or equal to A). Thus, Collatz process does not diverge to positive infinity and eventually reaches one digit in binary. Since one digit obtained from Collatz process in binary is equal to 1 in decimal, number of times that the Collatz process reaches 1 is limited.

We also prove that the reason why the number converged to 1 by repeating the Collatz process due to the fact that there is no multiple of 3 in the number after the Collatz process.

In graph theory, Collatz process is a directed graph and there is no closed path in a directed graph from the Collatz process. It means that all positive odd other than one do not enter an infinite loop. Therefore, we clarify that Collatz conjecture is positive.