Number Theory

In this course, integers are studied with little use of techniques from other mathematical fields. Questions of divisibility, use of the Euclidean algorithm to compute greatest common divisors, factorization of integers into prime numbers, investigation of perfect numbers and congruences belong here. Some important discoveries of this field are Fermat's little theorem, Euler's theorem, and the Chinese remainder theorem. The properties of multiplicative functions such as the Möbius function,  and Euler's φ function also fall into this area.  The course will take the student through questions in number theory that can be stated in elementary number theoretic terms, but  require very deep consideration and new approaches outside the realm of elementary number theory to solve. Examples include:

• The Goldbach conjecture concerning the expression of even numbers as sums of two primes.
• The twin prime conjecture about the infinitude of prime pairs.
• Fermat's last theorem (stated in 1637 but not proved until 1994) concerning the impossibility of finding nonzero integers x, y, z such that xn + yn = zn for some integer n greater than 2.