### Number Theory

Number theory abounds in problems that are easy to state, yet difficult to solve. An example is "Fermat's Last Theorem," stated by Pierre de Fermat about 350 years ago. Finding a proof of this theorem resisted the efforts of many mathematicians who developed new techniques in number theory, for example with the theory of elliptic curves over finite fields. A proof of Fermat's Last Theorem was finally presented by Andrew Wiles in 1995 in a landmark paper in the Annals of Mathematics.

Another famous problem from number theory is the Riemann hypothesis. This problem asks for properties of the Riemann zeta function, a function which plays a fundamental role in the distribution of prime numbers. Although it is over one hundred years old the Riemann hypothesis is still unresolved; in fact, the Clay Mathematics Institute has offered a prize of one million dollars for its solution.

Yet another famous open problem from number theory is the Goldbach conjecture which states that every even positive integer is a sum of two primes. Understanding this conjecture requires nothing more than high school mathematics, yet it has resisted the efforts of countless mathematicians.

#### Faculty

**Adebisi Agboola****PhD:**Columbia University, 1991**Interests:**Number Theory**Office:**Room 6724**email:**agboola@math.ucsb.edu

**Francesc Castella****PhD:**McGill University, 2013**Interests:**Number Theory and Arithmetic Geometry**Office:**Room 6511**email**: castella@math.ucsb.edu

**Jeffrey Stopple****PhD:**University of California, San Diego, 1986**Interests:**Number Theory**Office:**Room 6524**email:**stopple@math.ucsb.edu

**Yitang Zhang****PhD:**Purdue University, 1991**Interests:**Analytic Number Theory**Office:**Room 6721**email:**yitang.zhang@math.ucsb.edu

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