I dont understand mod - Page 2

On August 31 2010 13:25 blankspace wrote:
1. don't you mean p divides a? 1 + 1/2 + 1/3 + 1/4 = 25/12 for example.
Pair up the sum into pairs 1/x + 1/(p-x) = p/[x(p-x)]. If we group up stuff we get p(x/y) = a/b or
pxb = ay. p cannot divide y because the lcm of (1,2...p-1) does not contain p so p divides a.

On August 31 2010 10:26 GrayArea wrote:
What Math is this from? I have taken up to Calculus 2, and I've never seen "mod" before.

On August 31 2010 10:33 mieda wrote:


You can first get introduced to "mod" stuff for integers in any intro text in *elementary* number theory (as opposed to modern number theory, which uses just about every branch of mathematics including algebraic geometry, algebraic topology (and topology ofc), combinatorics, etc..). The "mod" stuff for general algebraic structures can be found in any intro to abstract algebra texts (such as Dummit & Foote, Lang, Hungerford, etc..).

For the elementary number theory part, there are many good texts that I used when I was in high school (while preparing for IMO, I made to MOSP and did well enough to be in black team but they wouldn't let me progress further because I did not have U.S. Citizenship at the time, much less a green card T_T) such as Ivan & Zuckermann (in fact this is the only intro elementary number theory book I read cover to cover and is said to be the "canonical" choice!)

Here's one of the books that I read cover to cover in high school which inspired me to major in math in college: http://www.amazon.com/Introduction-Theory-Numbers-Ivan-Niven/dp/0471625469/ref=sr_1_1?ie=UTF8&s=books&qid=1283218586&sr=8-1

It also has many excellent problems, some of which were proposed for IMO longlists (back in the days) I believe.

With that said, this is still elementary part of number theory. That is to say, classical 19th century stuff. Nowadays, thanks to Wiles and Taylor (who happens to be my advisor), modern number theory looks more like a fusion of algebraic geometry (with Grothendieck revolution - schemy stuff, for those of you who know what schemes / sheaf cohomology are ^^) and algebraic number theory mixed with galois representations, modular (and more generally automorphic) forms and representations.

If you'd like a little taste of how number theory has developed into in the last 100 years or so, Mazur has written up a nice article titled "Number as Gadfly"

Here's the link (in my dropbox): http://dl.dropbox.com/u/3799589/MazurGadfly.pdf