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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
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2 Answers

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, if you like an answer from an amateur...My 2 cents....

The only (in my opinion: remarkable) partial result is the disprove of whole classes of cycles (the so-called "1-cycles" and "m-cycles"). That means: independent of a specific length certain types of cycles were proven to be impossible first by Ray Steiner (1978(?)), later by John Simons and Benne de Weger (2000,2002,2006, see the wikipedia-entry at m-cycles cannot occur) (perhaps a similar situation like that of Kummer with Fermat's last conjecture, where he could prove FLT for complete classes of prime-exponents - but someone more educated than me might correct/improve that statement).

That successes were possible through results of the theory of rational approximation to transcendent numbers, namely linear forms of logarithms, here of log(2) and log(3). There is something more lingering there around, for instance a connection to a detail in the Waring-problem (see for instance mathworld, "power fractional parts") - so I recommend to take a deep look into that subject.

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Thanks for the answer. However, I was actually looking for an even earlier start than the m-cycles, transcendent numbers, power fractional parts, etc. I'm still lost and wondering actually, for someone with the usual maths you might study for a B.S. in engineering, physics, or applied math, what should I read next? I'm guessing something in Number Theory, but not sure where to start? –  Brett Pontarelli Jul 30 '12 at 20:12
    
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
    
Alternatively you might mail me privately and I could send you my set of articles which I collected over 2002-2009 from internet-discussions and also links to digitized versions of journal-articles which I've found in online-libraries in that time. –  Gottfried Helms Jul 30 '12 at 20:29
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