Sums of two cubes as twisted perfect powers, revisited

Abstract

In this paper, we sharpen earlier work of the first author, Luca and Mulholland, showing that the Diophantine equation $$A^3+B^3=q^{\alpha }{C}^p,ABC\ne 0,\gcd (A,B)=1,$$ 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 {\it 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 $p$.