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.
Faltings, Endlichkeitss¨ atze f¨ ur abelsche Variet¨ aten ¨ uber Zahlk¨ orpern,Invent
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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.