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arXiv · 2109.09128

Differences between perfect powers : the Lebesgue-Nagell Equation

Michael A. Bennett·Samir Siksek
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Abstract

We develop a variety of new techniques to treat Diophantine equations of the shape $x^2+D =y^n$, based upon bounds for linear forms in $p$-adic and complex logarithms, the modularity of Galois representations attached to Frey-Hellegouarch elliptic curves, and machinery from Diophantine approximation. We use these to explicitly determine the set of all coprime integers $x$ and $y$, and $n \geq 3$, with the property that $y^n > x^2$ and $x^2-y^n$ has no prime divisor exceeding $11$.

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Michael A. Bennett, Samir Siksek. 2021-09-19. Differences between perfect powers : the Lebesgue-Nagell Equation. https://arxiv.org/abs/2109.09128

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