Number Theory:

Number theory is the study of positive integers and modular arithmetic ("clock arithmetic" is modular arithmetic with the base 12).

You have already learned some important theorems of number theory -- for example, that there are an infinite number of primes, and that every positive integer may be factored into a product of primes, and this factorization is unique aside from the order. One of the most famous unsolved problems in mathematics comes from number theory. In geometry you saw "pythagorean triples" -- numbers like 3, 4, 5 where

The question is, can similar triples be found for an exponent higher than 2? "Fermat's Last Theorem" says that for n >2 it is not possible to find x, y, z all non-zero with

This is called "Fermat's Last Theorem" because he wrote in the margin of a book that he had a very interesting proof of this -- and shortly thereafter he died. Mathematicians have been trying to find his proof for the last 300 years!