# Sums of two cubes as twisted perfect powers, revisited

## TL;DR

This paper investigates the solutions of a specific cubic Diophantine equation involving prime powers, demonstrating that for most primes and large exponents, solutions are nonexistent, thus advancing understanding of sums of two cubes in number theory.

## Contribution

The paper improves previous results by showing nonexistence of solutions for most primes and large exponents, and introduces symplectic criteria to extend these results to more cases.

## Key findings

- Most primes q up to x have no solutions for the equation with large prime p.
- Conditional results show a positive proportion of prime exponents p yield no solutions.
- The work extends known families where solutions are proven not to exist.

## 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 $p$.