Sums of two cubes as twisted perfect powers, revisited

has, for “most” primes $q$ and suitably large prime exponents $p$ , no solutions. We handle a number of (presumably infinite) families where no such conclusion was hitherto known. Through further application of certain symplectic criteria, we are able to make some conditional statements about still more values of $q$ ; a sample such result is that, for all but $O (\sqrt{x} ∕ log x)$ primes $q$ up to $x$ , the equation

A^{3} + B^{3} = q C^{p} .

has no solutions in coprime, nonzero integers $A$ , $B$ and $C$ , for a positive proportion of prime exponents $p$ .

Keywords

Frey curves, ternary Diophantine equations, symplectic criteria

Mathematical Subject Classification 2010

Primary: 11D41

Milestones

Received: 24 February 2017

Revised: 7 September 2017

Accepted: 18 December 2017

Published: 11 July 2018

Authors

Michael A. Bennett
Department of Mathematics University of British Columbia Vancouver BC Canada

Carmen Bruni
Centre for Education in Mathematics and Computing University of Waterloo Waterloo ON Canada

Nuno Freitas
Department of Mathematics University of British Columbia Vancouver BC Canada

Vol. 12, No. 4, 2018

Michael A. Bennett, Carmen Bruni and Nuno Freitas

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors