Sums of two cubes as twisted perfect powers, revisited

Abstract

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

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

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 $p$.