Proof that Collatz conjecture is positive using the binary and the general term of progression of differences

Abstract

This paper presents a new proof of Collatz conjecture using the binary and the general term of progression of differences. 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 and the general term of progression of differences are used to prove that all positive odd otherthanonedonotenteraninfiniteloop(e.g. 1→3→4→1)by Collatz process. Using the 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. Therefore, we clarify that Collatz conjecture is positive.