Philadelphia Area Number Theory Seminar

Wednesday, July 11, 2018 - 2:15pm

Byungchul Cha

Muhlenberg College

Call (p, q) a Pythagorean pair if p and q are positive coprime integers such that p2 + q2 is a perfect square. Draw a line  from the origin into the first quadrant of the xy-plane. Suppose we want  to avoid all but finitely many Pythagorean pairs with as large a margin as possible. Which  is best? Which is second best? In this talk, we will introduce a dynamical system originally defined by Romik in 2008, study its Lagrange spectrum, and explain how our results can answer the above questions. We will review some well-known results on simple continued fractions and their Lagrange and Markoff spectra. This will provide a classical counterpart to our theory on Romiks dynamical system. This is joint work with Dong Han Kim (Dongguk University, South Korea). A large part of our talk will be accessible to undergraduates.