Ever wondered how large the jungle of Mathematics is? We'll I always loose my way in it, when I try to read through a paper. Mathematics, to me is the plain simple truth, that is put in abstract terms. You get reward points for being terse. It's elegant when you write a premise in one single step. I've been thinking about coming up with a list of simple theorems, I want these to serve as an aid for mapping the more inner paths of the Mathematics jungle.
I have been trying to study discrete mathematics for a while now. Here is one of the most important theorems (see The Art of Computer Programming, Volume 1, Page 41, Third Edition, D.E.Knuth) . The theorem is called Fermat's Theorem and states that
If p is a prime number then
For a good proof, see Knuth's book or Fermat's Little Theorem at Wolfram MathWorld
One of the most interesting applications of this theorem is the RSA algorithm. RSA is used quite extensively in public/private key based cryptosystems.
Checkout the RSA article on Wikipedia