Felipe Gonçalves


The Collatz Conjecture - V5B7 - Advanced Topics in Analysis - Master Program in Mathematics - Universität Bonn
Wednesdays and Fridays from 10 (c.t.) to 12 in Raum 0.003, Endenicher Allee 60.

Course Description

The idea of the course is simple: I will read a bunch of interesting papers about The 3x+1 Problem (The Collatz Conjecture), try to understand them to the best of my abilities, make notes and videos about it and make these available to you.
I am still figuring out what to do in the oral examinations.
The videos will be posted in my YouTube page: [Bomb Detonator]
The audio, resolution and illumination are not ideal but I am working to make it better.
I will publish new videos most likely twice a week, randomly. The goal is to make a total of something around 25 hours of lectures.
My notes are hand written, with several typos and maybe some minor mathematical mistakes.
Use my notes wisely --> [Click if you wanna to go to Narnia - VERSION 7]
Numerical Experiments for Mathematica --> [Life is happens to you when you're busy making other plans]

Preliminary List of Papers (not in order):


1*) J. Leslie Davison (1977), Some Comments on an Iteration Problem, Proc. 6-th Manitoba Conf. On Numerical Mathematics, and Computing (Univ. of Manitoba-Winnipeg 1976), Congressus Numerantium XVIII, Utilitas Math.: Winnipeg, Manitoba 1977, pp. 55–59. (MR 58 #31773).

2*) Ray P. Steiner (1978), A Theorem on the Syracuse Problem, Proc. 7-th Manitoba Con- ference on Numerical Mathematics and Computing (Univ. Manitoba-Winnipeg 1977), Congressus Numerantium XX, Utilitas Math.: Winnipeg, Manitoba 1978, pp. 553–559. (MR 80g:10003).

####2-adic extensions####

3) Daniel J. Bernstein (1994), A Non-Iterative 2-adic Statement of the 3x + 1 Conjecture, Proc. Amer. Math. Soc., 121 (1994), 405–408. (MR 94h:11108).

4) Daniel J. Bernstein and Jeffrey C. Lagarias (1996), The 3x + 1 Conjugacy Map, Canadian J. Math., 48 (1996), 1154-1169. (MR 98a:11027).

5*) Ethan Akin (2004), Why is the 3x + 1 Problem Hard?, In: Chapel Hill Ergodic Theory Workshops (I. Assani, Ed.), Contemp. Math. vol 356, Amer. Math. Soc. 2004, pp. 1–20. (MR 2005f:37031).

####dynamics approach####

6*) Marc Chamberland (1996), A Continuous Extension of the 3x + 1 Problem to the Real Line, Dynamics of Continuous, Discrete and Impulsive Dynamical Systems, 2 (1996), 495–509. (MR 97f:39031).

7) Vitaly Bergelson, Michal Misiurewicz and Samuel Senti (2006), Affine Actions of a Free Semigroup on the Real Line, Ergodic Theory and Dynamical Systems 26 (2006), 1285– 1305. MR2266362 (2008f:37019).

####probabilist apporach####

8) Konstantin Borovkov and Dietmar Pfeifer (2000), Estimates for the Syracuse problem via a probabilistic model, Theory of Probability and its Applications 45, No. 2 (2000), 300–310. (MR 1 967 765).

9) Stanislav Volkov (2006), A probabilistic model for the 5k + 1 problem and related prob- lems, Stochastic Processes and Applications 116 (2006), 662–674.

10) Jeffrey C. Lagarias and K. Soundararajan (2006), Benford’s Law for the 3x + 1 Function, J. London Math. Soc. 74 (2006), 289–303. (MR 2007h:37007)


11) R. E. Crandall, On the "3x + 1" problem, Math. Comp., 32 (1978) 1281-1292.

12) R. Terras, A stopping time problem on the positive integers, Acta Arith. 30 (1976) 241-252.

13) I. Krasikov, How many numbers satisfy the 3x+1 problem?,  International Journal of Mathematics and MAthematical Sciences 12(4).

14)  Ilia Krasikov and Jeffrey C. Lagarias, Bounds for the 3x + 1 problem using difference inequalities, Acta Arithmetica (2003), Volume 109, Issue:3, 237-258.

15) Terence Tao, Almost all orbits of the Collatz map attain almost bounded values.