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# Finding a rational point on the elliptic curve $$y^2 = x^3 + 7823$$

## Jennifer S. Balakrishnan

Week 2, Tuesday 5 May TT 2015
8:15pm

An elliptic curve E can be thought of as a smooth curve of the form

together with a special point $$O$$ ‘at infinity’. Suppose we fix integers $$A,B$$. The set of solutions in rational numbers to $$(*)$$, together with $$O$$, has the structure of an abelian group, and understanding this group $$E(Q)$$ has led to many interesting developments in number theory.

I’ll give an overview of the theory of ellipticcurves over the rationals, say a little about a million-dollar problem involving elliptic curves, and tell you about finding a rational point on the curve