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Twists of \(X(7)\) and primitive solutions to \(x^2+y^3=z^7\). (English) Zbl 1124.11019

Authors’ abstract: We find the primitive integer solutions to \(x^{2} +y^3=z^7\). A nonabelian descent argument involving the simple group of order 168 reduces the problem to the determination of the set of rational points on a finite set of twists of the Klein quartic curve \(X\). To restrict the set of relevant twists, we exploit the isomorphism between \(X\) and the modular curve \(X(7)\) and use modularity of elliptic curves and level lowering. This leaves 10 genus 3 curves, whose rational points are found by a combination of methods

MSC:

11D41 Higher degree equations; Fermat’s equation
11G10 Abelian varieties of dimension \(> 1\)
11G18 Arithmetic aspects of modular and Shimura varieties
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
14G05 Rational points

Software:

ecdata; Magma
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Online Encyclopedia of Integer Sequences:

Numbers n such that n^7 = a^2 + b^3 for positive integers a and b.

References:

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