Two and three descent for elliptic curves associated with perfect cuboids
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A rational perfect cuboid is a rectangular parallelepiped whose edges and face diagonals are given by rational numbers and whose space diagonal is equal to unity. Recently it was shown that the Diophantine equations describing such a cuboid lead to a couple of parametric families of elliptic curves. Two and three descent methods for calculating their ranks are discussed in the present paper. The elliptic curves in each parametric family are subdivided into two subsets admitting 2-descent and 3-descent methods respectively.
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Quartic reductions and elliptic obstructions for perfect Euler bricks
The perfect cuboid problem is equivalent to finding points on the curves w² = λ⁸ + Aλ⁴ + 1 with new elliptic obstructions excluding some families but no unconditional non-existence proof.
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