Number Theory Seminar

Keith ConradUniversity of Connecticut
Hall's conjecture and the ABC conjecture

Friday, September 28, 2018 - 2:25pm
Malott 205

Hall's conjecture is a statement about bounds on integral solutions $(x,y)$ to $y^2 = x^3+k$ where $k$ is a nonzero integer. It is known to be a consequence of the ABC conjecture, and it's natural to ask if the converse is true: does Hall's conjecture imply the ABC conjecture? In the literature there are equivalences of the ABC conjecture with variations on Hall's conjecture involving somewhat inelegant hypotheses: bounds when $(x,y)$ is 1 or 3, bounds on relatively prime solutions to both $y^2 = x^3+k$ and $3y^2 = x^3 + k$, and so on. We will give a sharper result: the ABC conjecture is equivalent to a bound on relatively prime solutions to $y^2 = x^3 + k$.