On $A^ 4+B^ 4+C^ 4=D^ 4$
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 by Noam D. Elkies PDF
 Math. Comp. 51 (1988), 825835 Request permission
Abstract:
We use elliptic curves to find infinitely many solutions to ${A^4} + {B^4} + {C^4} = {D^4}$ in coprime natural numbers A, B, C, and D, starting with \[ {2682440^4} + {15365639^4} + {18796760^4} = {20615673^4}.\] We thus disprove the $n = 4$ case of Euler’s conjectured generalization of Fermat’s Last Theorem. We further show that the corresponding rational points $( \pm A/D, \pm B/D, \pm C/D)$ on the surface ${r^4} + {s^4} + {t^4} = 1$ are dense in the real locus. We also discuss the smallest solution, found subsequently by Roger Frye.References

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Additional Information
 © Copyright 1988 American Mathematical Society
 Journal: Math. Comp. 51 (1988), 825835
 MSC: Primary 11D25; Secondary 11G35, 11G40
 DOI: https://doi.org/10.1090/S00255718198809302249
 MathSciNet review: 930224