Algebra & Number Theory

Sums of two cubes as twisted perfect powers, revisited

Michael A. Bennett, Carmen Bruni, and Nuno Freitas

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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

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

Received: 24 February 2017
Revised: 7 September 2017
Accepted: 18 December 2017
First available in Project Euclid: 28 July 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Frey curves ternary Diophantine equations symplectic criteria


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.

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