Natanael Wildner Fraga

math.NT — Number Theory

Rational Fourth Powers Summing to 82

Contributed by Natanael Wildner Fraga

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 determine the quotient modulo 2, prove that the displayed subgroup has odd index, and give the vanishing of the 2-primary part of the Tate–Shafarevich group. Two reductions determine the torsion. The finite arithmetic certificates are supplied in two self-contained Python programs; no height cutoff or conjectural rank computation is used.