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arXiv · 1702.07827

Sums of two cubes as twisted perfect powers, revisited

Michael A. Bennett·Carmen Bruni·Nuno Freitas
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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 $p$.

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Michael A. Bennett, Carmen Bruni, Nuno Freitas. 2017-02-25. Sums of two cubes as twisted perfect powers, revisited. https://arxiv.org/abs/1702.07827

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