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Abstract: We say n is a congruent
number if it is the area of a right triangle with sides all rational
numbers. Can we find an effective algorithm that will tell us when
n is congruent? A very nice characterization of such numbers was proven
by J. Tunnell in 1983. While the problem is classical and easy to
describe, Tunnell's theorem relied upon modern mathematical ideas
such as the relationship between elliptic curves and modular forms,
and the Burch-Swinnerton-Dyer conjecture. I will talk about the congruent
number problem and its relation to these topics. |