Number theory the branch of mathematics concerned with the
Natural Numbers
1,2,3,4,5, etc. Number theory has been studied from earliest times.
What makes number theory so cool is that propositions can be stated in a way
a grade schooler can understand, but the proofs often require the most subtile
insight and genuis.

Perhaps the most famous is Fermat's Last Theorem which
was proposed by
Pierre de Fermat around 1630 and was only recently
proved by Andrew Wiles in
1994. Fermat stated that he had a truly wonderous proof that the
equation has no solutions in positive integers ,
, and for any exponent , but that the
margin of the book was too small to hold it. We can't know if he had a valid
proof, but since the mathematics required to recently prove it were far more advanced
than what Fermat has available to him at the time, people highly doubt it.

Another example of a number theory conjecture that is simple to state is the
Goldbach Conjecture -- that every even integer greater
than 2 is the sum
of two primes. For example and . Stated by
Christian Goldback in 1742, it has been verified up
through 100,000, but has yet
to be proven.

Another incredibly cool thing about number theory is that it has practical uses.
One of the most famous examples of this is
RSA Cryptography developed in 1977 by Ronald Rivest, Adi Shamir,
and Len Adelman. RSA is a Public-Key Cryptography System
where the encryption
key and decryption key are different, so the encryption key can be told to everybody
who wants to send informtaion to the person with the secret decryption key without
fear of any evesdroppers being able to decrypt the message even if they have the
public key. RSA is used, for example, whenever your web browser goes into secure-mode.
The security of RSA lies in the difficulty of factoring huge numbers in any reasonable
amount of time.

Number Theory is one of the easiest branches of mathematics to state axioms and
derive formal proofs for. This has made it the subject of intense study as a formal
system which has led to interesting results like
Goedel's Incompleteness Theorem
which says that in any formal system powerful enough to include a description of
the integers, there exist statements which can neither be proven true nor false.
For an excelent and enjoyable descripion of some of Goedel's Incompleteness Theorem,
Number Theory in general, and a whole lot more,
I recomment Douglas Hofstadler's Goedel Escher Bach

On a personal note, I've always thought number theory was incredibly cool, but
could never fit a formal class on it into my schedule. My desire to sit down and
really learn it well combined with the lack of a need for complicated pictures
made number theory the first topic I attempted to cover on BrainFlux in any depth.
If this paragraph hasn't scared you off, I hope you learn something from all
of my efforts. :-)