The Diophantine Equation $x^ 4 + 2 y^ 4 = z^ 4 + 4 w^ 4$
HTML articles powered by AMS MathViewer
- by Andreas-Stephan Elsenhans and Jörg Jahnel PDF
- Math. Comp. 75 (2006), 935-940 Request permission
Abstract:
We show that, within the hypercube $|x|,|y|,|z|,|w| \leq 2.5 \cdot 10^6$, the Diophantine equation $x^4 + 2 y^4 = z^4 + 4 w^4$ admits essentially one and only one nontrivial solution, namely $(\pm 1\,484\,801, \pm 1\,203\,120, \pm 1\,169\,407, \pm 1\,157\,520)$. The investigation is based on a systematic search by computer.References
- Arnaud Beauville, Complex algebraic surfaces, London Mathematical Society Lecture Note Series, vol. 68, Cambridge University Press, Cambridge, 1983. Translated from the French by R. Barlow, N. I. Shepherd-Barron and M. Reid. MR 732439
- Daniel J. Bernstein, Enumerating solutions to $p(a)+q(b)=r(c)+s(d)$, Math. Comp. 70 (2001), no. 233, 389–394. MR 1709145, DOI 10.1090/S0025-5718-00-01219-9
- Thomas H. Cormen, Charles E. Leiserson, and Ronald L. Rivest, Introduction to algorithms, The MIT Electrical Engineering and Computer Science Series, MIT Press, Cambridge, MA; McGraw-Hill Book Co., New York, 1990. MR 1066870
- Pierre Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307 (French). MR 340258, DOI 10.1007/BF02684373
- Forster, O.: Algorithmische Zahlentheorie, Vieweg, Braunschweig 1996.
- Bjorn Poonen and Yuri Tschinkel (eds.), Arithmetic of higher-dimensional algebraic varieties, Progress in Mathematics, vol. 226, Birkhäuser Boston, Inc., Boston, MA, 2004. MR 2028897, DOI 10.1007/978-0-8176-8170-8
- Robert Sedgewick, Algorithms, Addison-Wesley Series in Computer Science, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1983. MR 784432
- Nigel P. Smart, The algorithmic resolution of Diophantine equations, London Mathematical Society Student Texts, vol. 41, Cambridge University Press, Cambridge, 1998. MR 1689189, DOI 10.1017/CBO9781107359994
- Swinnerton-Dyer, Sir P.: Rational points on fibered surfaces, in: Tschinkel, Y. (ed.): Mathematisches Institut, Seminars 2004, Universitätsverlag, Göttingen 2004, 103–109.
- André Weil, Sur les courbes algébriques et les variétés qui s’en déduisent, Publ. Inst. Math. Univ. Strasbourg, vol. 7, Hermann & Cie, Paris, 1948 (French). MR 0027151
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
- Received by editor(s): January 25, 2005
- Published electronically: 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 Gauß Laboratory 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 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 75 (2006), 935-940
- MSC (2000): Primary 11Y50; Secondary 14G05, 14J28
- DOI: https://doi.org/10.1090/S0025-5718-05-01805-3
- MathSciNet review: 2197001