2019/09/29 by Li, Longjiang
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1909.13218
This paper proposes a formula expression for the well-known Collatz conjecture (or 3x+1 problem), which can pinpoint all the growth points in the orbits of the Collatz map for any natural numbers. The Collatz map Col: N+1 → N+1 on the positive integers is defined as xn+1=Col(xn)=(3 xn +1)/2mn where xn+1 is always odd and mn is the step size required to eliminate any possible even values. The Collatz orbit for any positive integer, x1, is expressed by a sequence, and xn is defined as a growth point if Col(xn)>xn holds, and we show that every growth point is in a format of ``4y+3'' where y is any natural number. Moreover, we derive that, for any given positive integer n, there always exists a natural number, x1, that starts a monotonic increasing or decreasing Collatz sequence of length n with the same step size. For any given positive integer n, a class of orbits that share the same orbit rhythm of length n can also be determined.