Reduced Collatz Dynamics Data Reveals Properties for the Future Proof of Collatz Conjecture

Abstract

Collatz conjecture is also known as 3X + 1 conjecture. For verifying the conjecture, we designed an algorithm that can output reduced dynamics (occurred 3 × x+1 or x/2 computations from a starting integer to the first integer smaller than the starting integer) and original dynamics of integers (from a starting integer to 1). Especially, the starting integer has no upper bound. That is, extremely large integers with length of about 100,000 bits, e.g., 2100000 − 1, can be verified for Collatz conjecture, which is much larger than current upper bound (about 260). We analyze the properties of those data (e.g., reduced dynamics) and discover the following laws; reduced dynamics is periodic and the period is the length of its reduced dynamics; the count of x/2 equals to minimal integer that is not less than the count of (3 × x + 1)/2 times ln(1.5)/ln(2). Besides, we observe that all integers are partitioned regularly in half and half iteratively along with the prolonging of reduced dynamics, thus given a reduced dynamics we can compute a residue class that presents this reduced dynamics by a proposed algorithm. It creates one-to-one mapping between a reduced dynamics and a residue class. These observations from data can reveal the properties of reduced dynamics, which are proved mathematically in our other papers (see references). If it can be proved that every integer has reduced dynamics, then every integer will have original dynamics (i.e., Collatz conjecture will be true). The data set includes reduced dynamics of all odd positive integers in [3, 99999999] whose remainder is 3 when dividing 4, original dynamics of some extremely large integers, and all computer source codes in C that implement our proposed algorithms for generating data (i.e., reduced or original dynamics).