Title: 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 $$ A^3+B^3 = q^\alpha C^p, \,
\, ABC \neq 0, \, \, \gcd (A,B) =1, $$ 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 $p$.