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Mathematics of Computation

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On $ A\sp 4+B\sp 4+C\sp 4=D\sp 4$

Author: Noam D. Elkies
Journal: Math. Comp. 51 (1988), 825-835
MSC: Primary 11D25; Secondary 11G35, 11G40
MathSciNet review: 930224
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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

$\displaystyle {2682440^4} + {15365639^4} + {18796760^4} = {20615673^4}.$

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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Article copyright: © Copyright 1988 American Mathematical Society