## Algebra & Number Theory

### 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

$A 3 + B 3 = q α C p , A B C ≠ 0 , gcd ( A , B ) = 1 ,$

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 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 $O ( x ∕ log x )$ 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 $p$.

#### Article information

Source
Algebra Number Theory, Volume 12, Number 4 (2018), 959-999.

Dates
Revised: 7 September 2017
Accepted: 18 December 2017
First available in Project Euclid: 28 July 2018

https://projecteuclid.org/euclid.ant/1532743327

Digital Object Identifier
doi:10.2140/ant.2018.12.959

Mathematical Reviews number (MathSciNet)
MR3830208

Zentralblatt MATH identifier
06911691

Subjects
Primary: 11D41: Higher degree equations; Fermat's equation

#### Citation

Bennett, Michael A.; Bruni, Carmen; Freitas, Nuno. Sums of two cubes as twisted perfect powers, revisited. Algebra Number Theory 12 (2018), no. 4, 959--999. doi:10.2140/ant.2018.12.959. https://projecteuclid.org/euclid.ant/1532743327

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