[aoriste]

A trivial tangent: oddly enough I find that, whether I'm programming for distributed applications or animations, it is the number theory, and not (surprisingly) the linear algebra, that I am most grateful for having learned. Boutade's Law: do not underestimate the algorithmic utility of knowing your way around the integers.

[infinite8s]

This is intriguing. Care to elaborate on a problem you've had and how number theory helped? I did the typical math courses required for a computer science degree (discrete math, linear algebra, diffeq, probability and statistics), but stopped short of the serious math courses (analysis). I've been meaning to start up with analysis (mainly for fun) but I'm intrigued to see how it might help in my day to day work.

[aoriste]

Well, unless you're terribly interested in moving on to things like differential geometry, I'd put analysis off a bit longer were I you. If you'd like something stimulating and potentially useful, more advanced combinatorics, number theory, or algebraic geometry might all be good choices. The first two tend to overlap a bit when they are presented in textbooks (with some group theory tossed in as well). If you do crazy graphics programming, or if you program robots' spatial reasoning, algebraic geometry might pay off - though, AG doesn't have as easy an entry. (One exception might be via the book Computational Algebraic Geometry by Hal Schenck).

As far as number theory "paying off": 1) Many proofs in number theory are algorithmic in nature; 2) Computers understand integers with greater facility than they do the psuedo-reals we call floats - often times efficient integer approximations will be more appropriate than slower floating point solutions, and knowing the integers will help you develop/understand these approximations; 3) If cryptography is your bag, number theory is a must; 4) Martin Davis, Yuri Matiyasevich, Julia Robinson, Hilbert's 10th problem, Turing Machines, computability theory, and (the number theory bit) diophantine equations (sorry to be cryptic, this is getting rather long winded).

[zackattack]

I would love to hear some of the number theory gems that have been most useful for you.