Skip to main content

On the density of parameterizations of generalized Fermat equations of signature (2,3,3) that produce locally primitive solutions

Resource type
Thesis type
(Thesis) M.Sc.
Date created
We consider the equations Ax^2+By^3=Cz^3, where A,B,C are square-free and pairwise co-prime integers. A solution (x,y,z) is called primitive if it consists of co-prime integers. Adapting earlier work for the equations x^2+y^3=Cz^3, we show that primitive solutions give rise to integer Klein forms of degree four, with discriminant A^3B^2C . Whether Klein forms come from primitive solutions is determined by local conditions. We show that for primes p dividing B, there are exactly four GL_2(Q_p)-equivalence classes of Klein forms that are relevant, and that exactly half of those classes come from Z_p-primitive solutions. We also show that if we set A=1, then further restricting B,C to square-free and co-prime integers leaves us with an asymptotically positive proportion of triples.
Copyright statement
Copyright is held by the author.
This thesis may be printed or downloaded for non-commercial research and scholarly purposes.
Scholarly level
Supervisor or Senior Supervisor
Thesis advisor: Bruin, Nils
Member of collection

Views & downloads - as of June 2023

Views: 43
Downloads: 0