#### Vol. 12, No. 4, 2018

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors 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
##### Abstract

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

${A}^{3}+{B}^{3}={q}^{\alpha }{C}^{p},\phantom{\rule{1em}{0ex}}ABC\ne 0,\phantom{\rule{1em}{0ex}}gcd\left(A,B\right)=1,$

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 $O\left(\sqrt{x}∕logx\right)$ 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 $p$.

However, your active subscription may be available on Project Euclid at
https://projecteuclid.org/ant

We have not been able to recognize your IP address 3.235.228.219 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

or by using our contact form.