# Prerequisite reading before studying the Collatz $3x+1$ Problem

Let's assume I am starting college and have just finished calculus. I've been reading a bit online about the Collatz $3x+1$ Problem and find it to be very intriguing. However, a lot of what I'm reading uses terms and techniques that I have not seen before. I'm wondering what prerequisite (text book) reading is required before starting to study this problem?

Put another way: I'm thinking about reading The Ultimate Challenge: The 3x+1 Problem by Jeffrey C. Lagarias. What areas of mathematics will I need to understand first before being able to fully understand this book?

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Almost the same: math.stackexchange.com/questions/158291/… –  Zander Aug 7 '12 at 15:56
@Zander I think the difference is that in that question the OP claims he has "enough knowledge to work on a singular facet of the problem". Where as I am perhaps a step or two behind this person. I'll updated my post and see if I can clarify. –  Brett Pontarelli Aug 14 '12 at 22:25

The Lagarias book is a compilation of papers by various authors about various aspects of the problem. Different papers have different prerequisites. Some of the more expository papers have essentially no prerequisites at all; for others, you'll want to know about dynamical systems, Markov chains, ergodic theory, $p$-adic numbers, Turing machines and undecideability, and, of course, elementary Number Theory. And each of these has prerequisites, e.g., ergodic theory is based on measure theory, Markov chains involve Linear Algebra, etc., etc., etc. But don't be disheartened! You don't need all these for every paper, not by any means, and a well-written paper will teach you something useful in its introductory paragraphs even if the rest of the paper is beyond you.

I think the best thing is to jump in, start reading something you find interesting, and then, if you get stuck, come back here to ask something like, "What do I need to know to understand the proof that all furbles are craginacs, as given on page 977 of Peeble and Zimp, The Elephant and the $3x+1$ Problem?" It's much easier to give prerequisites when you have a narrowly-focussed problem in mind, than when it's as broad as "I want to learn about the $3x+1$ problem".

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Hmm, the path to the Steiner/Simons-approach begins with modular consideration. So the modularity of exponential forms, a simple example $\small 3^n-1$ divisible by 5 dependent on n etc was a good start for me. Long time I did not go into the problem deeply, so I do not know the currently best introductions. Perhaps my own two treatizes might be a door for the absolute beginner? Try go.helms-net.de and click one of the two links for the Collatz-problem. They also give further links (I remember "Ken Conrow", and others) –  Gottfried Helms Jul 30 '12 at 20:23