MTH 238: Topics in Number Theory

Probably no area of mathematics has been studied more for the sheer enjoyment of its problems and ideas than number theory, an area that considers questions related to the integers. One of its enticements is that many unsolved problems in number theory can be stated in very simple terms, yet attempts to solve them have led to some of the most significant work of modern mathematics. The most famous example of this is the work related to Fermat's last theorem, both the many attempts to prove it since it was announced over 350 years ago as well as the widely heralded 1994 proof by Wiles and Taylor. Fermat's last theorem simply states that for $n > 2$ there are no positive integers $x$, $y$, and $z$ which satisfy $x^n+y^n=z^n$.

This course surveys major techniques, problems, and accomplishments of the field. Topics include divisibility, prime numbers, the Euclidean algorithm, factorization methods, congruences, Fermat's little theorem, pseudoprimes, Euler's theorem, Mersenne primes, primitive roots, quadratic reciprocity, Diophantine equations, Fermat's last theorem, and modern applications to computer science and cryptology. Some of the unsolved problems discussed are:

1. The Twin Prime problem: There are many examples of pairs of primes that differ by two, such as 3 and 5, 5 and 7, 17 and 19. Are there infinitely many of these?
   
2. Are there infinitely many Mersenne primes? A Mersenne prime is a prime number of the form $2^n-1$, such as $2^2-1=3$, $2^3-1=7$, and $2^5-1=31$. They were discovered by Euclid in 350 BC. The largest known Mersenne prime (as of October 10, 2004) is $2^{24,036,583}-1$ that was found May 28, 2004.
   
3. The Goldbach Conjecture: Any even number $> 2$ may be expressed as the sum of two prime numbers. (Examples: $4=2+2, 10=3+7, 48=17+31$.)
   
4. An integer is said to be perfect if it is the sum of its positive divisors other than itself. An example is $6=1+2+3$. Are there infinitely many perfect numbers? Are there any odd perfect numbers?