Rational Fourth Powers Summing to 82
We prove that the only primitive positive integer solutions of $x^4+y^4=82z^4$ are $(1,3,1)$ and $(3,1,1)$. An arithmetic reduction lifts every other putative solution to an elliptic curve over $K=\mathbb{Q}(\sqrt{41},\sqrt{2})$. An explicit 2-descent determines its rank as 3. A logarithmic functional then excludes all nonzero parameters in the relevant 2-adic disc, including both relative sign patterns at the two places above 2. The same argument determines every point of the auxiliary quartic whose parameter is rational: only the two points at infinity occur. Exact square identities also…
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