# y^2 = x^3 + Ax + B

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Levent Alpoge, Princeton University
Fine Hall 322

Solving the titular equation in integers x and y is a problem that goes back to Diophantus himself in his famous book on, yes, Diophantine equations. Bhargava and collaborators have recently made progress on the expectation that, when sampling over random A and B, one should expect no *rational* solutions half the time, and infinitely many (but minimally so, in a sense) rational solutions the other half of the time. So what of integral solutions? Siegel showed long ago that there are always finitely many integer solutions to such an equation (when the discriminant is nonzero). In this talk I will show that there are at most boundedly many (< 66) on average.