Abstract: Examples of Pythagorean triplets,
3
2 + 42 = 52
, 5
2 + 122 = 132
, 242 + 72 = 252
, · · ·
have been known since the time of the ancient Sumerians. Starting with the theorem of Pythagoras and a beautiful proof attributed to Baudhayana (200 years before Pythagoras), we will describe the general
formula to get all Pythagorean triplets.
We will next discuss how to solve more general quadratic equations using geometry, making use of stereographic projections. We will also relate it to the famous t = tan(θ/2) substitution used in integrating trignometric functions.
This leads us to a theorem of Legendre and the beginnings of the localglobal principle in number theory. We conclude by stating some open questions. The talk should be accessible to advanced school students.