Vol. 12, No. 4, 2018

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Sums of two cubes as twisted perfect powers, revisited

Michael A. Bennett, Carmen Bruni and Nuno Freitas

Vol. 12 (2018), No. 4, 959–999
Abstract

We sharpen earlier work (2011) of the first author, Luca and Mulholland, showing that the Diophantine equation

A3 + B3 = qαCp,ABC0,gcd(A,B) = 1,

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(xlogx) primes q up to x, the equation

A3 + B3 = qCp.

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