Brian Conrad
University of Michigan

Tuesday, March 9
3:00 p.m.
New Science Facility
 

Conics via algebra and geometry

 One of the most beautiful theorems in elementary mathematics is the formula that parameterizes all Pythagorean triples. This is a formula that can be proved by the methods of algebra, but such a proof sheds no light on how one might have discovered the formula, or the range of situations for which one may expect such formulas to exist.  For example, if we insert nontrivial coefficients on the Pythagorean equation, can we still expect to find a parametric formula for all of its solutions?

These matters are all immensely clarified by viewing the problem from the point of view of geometry, not algebra.  In fact, the parametric formula for the solutions to the Pythagorean equation rests on a special case of a general geometric property of conics in the plane.  We will present this geometric argument, explain how it yields formulas that solve the Pythagorean equation as well as many others, and the geometry will lead us to raise a fundamental problem in arithmetic that was first solved by Gauss in his book Disquisitiones Arithmeticae that created modern number theory.