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The Diophantine Equation $ x\sp4 + 2 y\sp4 = z\sp4 + 4 w\sp4$

Author(s): Andreas-Stephan Elsenhans; Jörg Jahnel.
Journal: Math. Comp. 75 (2006), 935-940.
MSC (2000): Primary 11Y50; Secondary 14G05, 14J28
Posted: December 19, 2005
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Abstract: We show that, within the hypercube $ \vert x\vert,\vert y\vert,\vert z\vert,\vert w\vert \leq 2.5 \cdot 10\sp6$, the Diophantine equation $ x\sp4 + 2 y\sp4 = z\sp4 + 4 w\sp4$ admits essentially one and only one nontrivial solution, namely $ (\pm1\,484\,801, \pm1\,203\,120, \pm1\,169\,407, \pm1\,157\,520)$. The investigation is based on a systematic search by computer.

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Additional Information:

Andreas-Stephan Elsenhans
Affiliation: Mathematisches Institut der Universität Göttingen, Bunsenstraße 3--5, D-37073 Göttingen, Germany
Email: elsenhan@uni-math.gwdg.de

Jörg Jahnel
Affiliation: Mathematisches Institut der Universität Göttingen, Bunsenstraße 3--5, D-37073 Göttingen, Germany
Email: jahnel@uni-math.gwdg.de

DOI: 10.1090/S0025-5718-05-01805-3
PII: S 0025-5718(05)01805-3
Keywords: $K3$~surface, diagonal quartic surface, rational point, Diophantine equation, computer solution, hashing
Received by editor(s): January 25, 2005
Posted: December 19, 2005
Additional Notes: The first author was partially supported by a Doctoral Fellowship of the Deutsche Forschungsgemeinschaft~(DFG)
The computer part of this work was executed on the Linux PCs of the GaußLaboratory for Scientific Computing at the Göttingen Mathematisches Institut. Both authors are grateful to Professor Y. Tschinkel for the permission to use these machines as well as to the system administrators for their support
Copyright of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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