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?
 A: 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". 
A: 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.
