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On $A^4 + B^4 + C^4 = D^4$

Noam D. Elkies
Mathematics of Computation
Vol. 51, No. 184 (Oct., 1988), pp. 825-835
DOI: 10.2307/2008781
Stable URL: http://www.jstor.org/stable/2008781
Page Count: 11
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On $A^4 + B^4 + C^4 = D^4$
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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.

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