The Collatz Problem generalized to 3x+k

2021/01/20 by Franz Wegner, Wegner, Franz
Mathematics · #Benford’s Law and Fraud Detection #Dynamical Systems (math.DS) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2101.08060

openalex publication_date 2021/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Collatz problem with 3x+k is revisited. Positive and negative limit cycles are given up to k=9997 starting with x₀=-2⋅10⁷...+2⋅10⁷. A simple relation between the probability distribution for the Syracuse iterates for various k (not divisible by 2 and 3) is obtained. From this it follows that the oscillation considered by Tao 2019 ( arXiv:1909.03562v2 ) does not depend on k. Thus this piece of the proof of his theorem 1.3 "Almost all Collatz orbits attain almost bounded values" holds for all k not divisible by 2 and 3.

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