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. :-)