Sums of two cubes as twisted perfect powers, revisited

2017/02/25 by Bennett, Michael A., Bruni, Carmen, Freitas, Nuno · 1 citation
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1702.07827

Abstract

In this paper, we sharpen earlier work of the first author, Luca and Mulholland, showing that the Diophantine equation A³+B³ = q^αC^p, ABC ≠ 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(√(x)/log x) primes q up to x, the equation A³ + B³ = 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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• Local points on twists of X(p) with applications

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