Vol. 12, No. 4, 2018

Download this article
Download this article For screen
For printing
Recent Issues

Volume 13
Issue 8, 1765–1981
Issue 7, 1509–1763
Issue 6, 1243–1507
Issue 5, 995–1242
Issue 4, 749–993
Issue 3, 531–747
Issue 2, 251–530
Issue 1, 1–249

Volume 12, 10 issues

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Editorial Board
Editors' Interests
Submission Guidelines
Submission Form
Editorial Login
Ethics Statement
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Author Index
To Appear
Other MSP Journals
Sums of two cubes as twisted perfect powers, revisited

Michael A. Bennett, Carmen Bruni and Nuno Freitas

Vol. 12 (2018), No. 4, 959–999

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.

Frey curves, ternary Diophantine equations, symplectic criteria
Mathematical Subject Classification 2010
Primary: 11D41
Received: 24 February 2017
Revised: 7 September 2017
Accepted: 18 December 2017
Published: 11 July 2018
Michael A. Bennett
Department of Mathematics
University of British Columbia
Vancouver BC
Carmen Bruni
Centre for Education in Mathematics and Computing
University of Waterloo
Waterloo ON
Nuno Freitas
Department of Mathematics
University of British Columbia
Vancouver BC