**Session leader**: Brian Conrad

**Session topic**: Pythagorean Triples and Conics

Among the solutions to the Pythagorean equation in whole numbers, some examples such as (3,4,5) and (5,12,13) are generally familiar (and perhaps (8,15,17) is to a lesser extent). Are there infinitely many such triples if we disregard the operation of scaling (e.g., creating (6,8,10) from (3,4,5))? The answer is affirmative in a definitive way: there is a nifty explicit formula that generates all such examples in terms of 2 parameters.

But much more remarkable than the purely algebraic fact of such a formula is that it has an elegant geometric explanation that is the tip of the iceberg on getting geometric insight into variations of the Pythagorean equation (involving ellipses and hyperbolas). This will provide a hands-on introduction to some basic ideas in the fascinating subject of algebraic geometry.