Differences between perfect powers: The Lebesgue-Nagell equation

Bennett, Michael; Siksek, Samir

doi:10.1090/tran/8734

Differences between perfect powers: The Lebesgue-Nagell equation
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Trans. Amer. Math. Soc. 376 (2023), 335-370 Request permission

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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