New integer representations as the sum of three cubes
Authors:
Michael Beck, Eric Pine, Wayne Tarrant and Kim Yarbrough Jensen
Journal:
Math. Comp. 76 (2007), 1683-1690
MSC (2000):
Primary 11D25; Secondary 11Y50, 11N36.
DOI:
https://doi.org/10.1090/S0025-5718-07-01947-3
Published electronically:
March 14, 2007
MathSciNet review:
2299795
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: We describe a new algorithm for finding integer solutions to $x^3 + y^3 + z^3 = k$ for specific values of $k$. We use this to find representations for values of $k$ for which no solution was previously known, including $k=30$ and $k=52$.
- Eric Bach and Jeffrey Shallit, Algorithmic number theory. Vol. 1, Foundations of Computing Series, MIT Press, Cambridge, MA, 1996. Efficient algorithms. MR 1406794
- Andrew Bremner, On sums of three cubes, Number theory (Halifax, NS, 1994) CMS Conf. Proc., vol. 15, Amer. Math. Soc., Providence, RI, 1995, pp. 87–91. MR 1353923, DOI https://doi.org/10.1006/jnth.1995.1021
- J. W. S. Cassels, A note on the Diophantine equation $x^3+y^3+z^3=3$, Math. Comp. 44 (1985), no. 169, 265–266. MR 771049, DOI https://doi.org/10.1090/S0025-5718-1985-0771049-4
- W. Conn and L. N. Vaserstein, On sums of three integral cubes, The Rademacher legacy to mathematics (University Park, PA, 1992) Contemp. Math., vol. 166, Amer. Math. Soc., Providence, RI, 1994, pp. 285–294. MR 1284068, DOI https://doi.org/10.1090/conm/166/01628
- Noam Elkies, $<$elkies@abel.math.harvard.edu$>$ “x^3 + y^3 + z^3 = d," 9 July 1996, $<$nmbrthry@listserv.nodak.edu$>$ via $<$http://listserv.nodak.edu/archives/nmbrthry.html$>$.
- V. L. Gardiner, R. B. Lazarus, and P. R. Stein, Solutions of the diophantine equation $x^{3}+y^{3}=z^{3}-d$, Math. Comp. 18 (1964), 408–413. MR 175843, DOI https://doi.org/10.1090/S0025-5718-1964-0175843-9
- D. R. Heath-Brown, Searching for solutions of $x^3+y^3+z^3=k$, Séminaire de Théorie des Nombres, Paris, 1989–90, Progr. Math., vol. 102, Birkhäuser Boston, Boston, MA, 1992, pp. 71–76. MR 1476729, DOI https://doi.org/10.1007/978-1-4757-4269-5_6
- D. R. Heath-Brown, W. M. Lioen, and H. J. J. te Riele, On solving the Diophantine equation $x^3+y^3+z^3=k$ on a vector computer, Math. Comp. 61 (1993), no. 203, 235–244. MR 1202610, DOI https://doi.org/10.1090/S0025-5718-1993-1202610-5
- Kenji Koyama, Tables of solutions of the Diophantine equation $x^3 + y^3 + z^3 = n$, Mathematics of Computation 62 (1994), 941–942.
- Kenji Koyama, Yukio Tsuruoka, and Hiroshi Sekigawa, On searching for solutions of the Diophantine equation $x^3+y^3+z^3=n$, Math. Comp. 66 (1997), no. 218, 841–851. MR 1401942, DOI https://doi.org/10.1090/S0025-5718-97-00830-2
- D. H. Lehmer, On the Diophantine equation $x^3+y^3+z^3=1$, J. London Math. Soc. 31 (1956), 275–280. MR 78397, DOI https://doi.org/10.1112/jlms/s1-31.3.275
- Richard F. Lukes, A Very Fast Electronic Number Sieve, University of Manitoba doctoral thesis, 1995.
- Kurt Mahler, Note On Hypothesis K of Hardy and Littlewood, Journal of the London Mathematical Society 11 (1936), 136–138.
- J. C. P. Miller and M. F. C. Woollett, Solutions of the Diophantine equation $x^3+y^3+z^3=k$, J. London Math. Soc. 30 (1955), 101–110. MR 67916, DOI https://doi.org/10.1112/jlms/s1-30.1.101
- L. J. Mordell, On sums of three cubes, J. London Math. Soc. 17 (1942), 139–144. MR 7761, DOI https://doi.org/10.1112/jlms/s1-17.3.139
- L. J. Mordell, On an infinity of integer solutions of $ax^3+ay^3+bz^3=bc^3$, J. London Math. Soc. 30 (1955), 111–113. MR 67917, DOI https://doi.org/10.1112/jlms/s1-30.1.111
- S. Ryley, The Ladies’ Diary 122 (1825), 35.
- Manny Scarowsky and Abraham Boyarsky, A note on the Diophantine equation $x^{n}+y^{n}+z^{n}=3$, Math. Comp. 42 (1984), no. 165, 235–237. MR 726000, DOI https://doi.org/10.1090/S0025-5718-1984-0726000-9
- Gérald Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, 2nd ed., Cours Spécialisés [Specialized Courses], vol. 1, Société Mathématique de France, Paris, 1995 (French). MR 1366197
- R. C. Vaughan, A new iterative method in Waring’s problem, Acta Math. 162 (1989), no. 1-2, 1–71. MR 981199, DOI https://doi.org/10.1007/BF02392834
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Additional Information
Michael Beck
Affiliation:
Department of Mathematics, University of Georgia, Athens, Georgia 30602
Email:
mbeck@math.uga.edu
Eric Pine
Affiliation:
Department of Mathematics, University of Georgia, Athens, Georgia 30602
Email:
epine@math.uga.edu
Wayne Tarrant
Affiliation:
Department of Mathematics, University of Georgia, Athens, Georgia 30602
Kim Yarbrough Jensen
Affiliation:
Department of Mathematics, University of Georgia, Athens, Georgia 30602
Received by editor(s):
February 7, 2002
Received by editor(s) in revised form:
October 8, 2005
Published electronically:
March 14, 2007
Article copyright:
© Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.