There are many beautiful identities involving positive integers. For example, Pythagoras knew 3^2 + 4^2 = 5^2 while Plato knew 3^3 + 4^3 + 5^3 = 6^3. Euler discovered 59^4 + 158^4 = 133^4 + 134^4, and even a famous story involving G. H. Hardy and Srinivasa Ramanujan involves 1^3 + 12^3 = 9^3 + 10^3. But how does one find such identities? Around the third century, the Greek mathematician Diophantus of Alexandria introduced a systematic study of integer solutions to polynomial equations. In this talk, we'll focus on various types of so-called Diophantine Equations, setting up the weekend's discussion of topics such as Pythagorean Triples, Pell's Equations, Elliptic Curves, and Fermat's Last Theorem.
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