Numbers form the basis of much of the mathematics we study in school, yet there are many unanswered questions about them. The most famous of these, on the validity of Fermat's Last Theorem, was finally answered only in the last decade after resisting three centuries of challenges from mathematicians. Many of these problems, and their solutions, are accessible to anyone who knows basic algebra.

This class will begin with straightforward algebra and progress to more difficult and more interesting problems. One problem is factoring: much of the security of the internet is based on the presumption that this is difficult. For example, RSA, whose encryption algorithm we will study, has cash prizes for factoring large numbers that are the product of two primes. (A whole number p>1 is prime if it is divisible only by 1 and itself.) A related problem is finding larger and larger prime numbers. (There are also prizes for computing sufficiently large primes.) For example, it is currently unknown whether or not the 32nd Fermat number F₃₂=2²³²+1 is prime.

Our approach is motivated by mathematics research. We will search for patterns in hands-on examples and discover (and prove) rules based on these patterns. The topic of number theory allows us to ground our abstract results in terms of straightforward examples; that is, our mathematically rigorous results will come from experimenting with shapes via problem-solving, games, and computer programs.