Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



New sums of three cubes

Authors: Andreas-Stephan Elsenhans and Jörg Jahnel
Journal: Math. Comp. 78 (2009), 1227-1230
MSC (2000): Primary 11Y50; Secondary 14G05, 14J28
Published electronically: August 20, 2008
MathSciNet review: 2476583
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We report on our search for solutions of the Diophantine equation $ x^3 + y^3 + z^3 = n$ for $ n < 1000$ and  $ \vert x\vert, \vert y\vert, \vert z\vert < 10^{14}$.

References [Enhancements On Off] (What's this?)

  • [AMD] Software Optimization Guide for AMD Athlon$ ^{\rm TM}$ 64 and AMD Opteron$ ^{\rm TM}$ Processors, Rev. 3.04, AMD, Sunnyvale, CA, 2004.
  • [BPTY] Beck, M., Pine, E., Tarrant, W., and Yarbrough Jensen, K.: New integer representations as the sum of three cubes, Math. Comp. 76 (2007), 1683-1690. MR 2299795 (2007m:11170)
  • [Be] Bernstein, D.: Threecubes, available at:
  • [El] Elkies, N. D.: Rational points near curves and small nonzero $ \vert x^3 - y^2\vert$ via lattice reduction, in: Algorithmic number theory (Leiden 2000), Lecture Notes in Computer Science 1838, Springer, Berlin 2000, 33-63. MR 1850598 (2002g:11035)
  • [EJ] Elsenhans, A.-S. and Jahnel, J.: List of solutions of $ x^3 + y^3 + z^3 = n$ for  $ n < 1\,000$ neither a cube nor twice a cube, available at: threecubes_20070419.txt.
  • [FP] Fincke, U. and Pohst, M.: Improved methods for calculating vectors of short length in a lattice, including a complexity analysis, Math. Comp. 44 (1985), 463-471. MR 777278 (86e:11050)
  • [HB] Heath-Brown, D. R.: The density of zeros of forms for which weak approximation fails, Math. Comp. 59 (1992), 613-623. MR 1146835 (93a:11055)
  • [HLR] Heath-Brown, D. R., Lioen, W. M., and te Riele, H. J. J.: On solving the diophantine equation $ x^3 + y^3 +z^3 = k$ on a vector computer, Math. Comp. 61 (1993), 235-244. MR 1202610 (94f:11132)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 11Y50, 14G05, 14J28

Retrieve articles in all journals with MSC (2000): 11Y50, 14G05, 14J28

Additional Information

Andreas-Stephan Elsenhans
Affiliation: Mathematisches Institut der Universität Göttingen, Bunsenstrasse 3–5, D-37073 Göttingen, Germany

Jörg Jahnel
Affiliation: Mathematisches Institut der Universität Göttingen, Bunsenstrasse 3–5, D-37073 Göttingen, Germany

Keywords: Diophantine equation, sum of three cubes, Elkies' method
Received by editor(s): February 12, 2008
Received by editor(s) in revised form: April 10, 2008
Published electronically: August 20, 2008
Additional Notes: The computer part of this work was executed on the Sun Fire V20z Servers of the Gauss Laboratory for Scientific Computing at the Göttingen Mathematisches Institut. Both authors are grateful to Professor Y. Tschinkel for permission to use these machines as well as to the system administrators for their support.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society