Generalized GCDs as Applications of Vojta’s Conjecture
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Date
2023-08-31
Authors
Pyott, Nolan
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Publisher
University of Waterloo
Abstract
Starting with an analysis of the result that for any coprime integers a and b, and some ϵ > 0, we have eventually that gcd(a^n − 1,b^n − 1) < a^ϵn holds for all n, we are motivated to look for geometric reasons why this should hold. After some discussion on the general geometry and arithmetic needed to examine these questions, we take a quick look into how Vojta’s conjectures provide a generalization of our first result. In particular, we also note a case where this implies a similar equality on particular elliptic curves.
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Keywords
diophantine geometry, Vojta's conjecture, algebraic geometry, arithmetic, gcd, elliptic curve, projective variety