Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



Ideal 9th-order multigrades and Letac's elliptic curve

Author: C. J. Smyth
Journal: Math. Comp. 57 (1991), 817-823
MSC: Primary 11D72; Secondary 11G05
MathSciNet review: 1094960
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: By showing that the elliptic curve $ ({x^2} - 13)({y^2} - 13) = 48$ has infinitely many rational points, we prove that Letac's construction produces infinitely many genuinely different ideal 9th-order multigrades. We give one (not very small) new example, and, by finding the Mordell-Weil group of the curve, show how to find all examples obtainable by Letac's method.

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

  • [1] J. W. S. Cassels, Diophantine equations with special reference to elliptic curves, J. London Math. Soc. 44 (1966), 193-291. MR 0199150 (33:7299)
  • [2] N. D. Elkies, On $ {A^4} + {B^4} + {C^4} = {D^4}$, Math. Comp. 51 (1988), 825-835. MR 930224 (89h:11012)
  • [3] W. Fulton, Algebraic curves, Benjamin, New York, 1969. MR 0313252 (47:1807)
  • [4] A. Gloden, Mehrgradige Gleichungen, Noordhoff, Groningen, 1944. MR 0019638 (8:441f)
  • [5] T. Nagell, Solution de quelque problèmes dans la théorie arithmétique des cubiques planes du premier genre, Vid. Akad. Skrifter Oslo 1 (1935), no. 1.
  • [6] E. Rees and C. Smyth, On the constant in the Tarry-Escott Problem, Fifty Years of Polynomials, Lecture Notes in Math., vol. 1415, Springer, Berlin and New York, 1990, pp. 196-208. MR 1044114 (91g:11030)
  • [7] J. H. Silverman, The arithmetic of elliptic curves, Springer, Berlin and New York, 1986. MR 817210 (87g:11070)
  • [8] E. M. Wright, personal communication, 1989.
  • [9] -, The Tarry-Escott and the "easier" Waring problems, J. Reine Angew. Math. 311/312 (1979), 170-173. MR 549963 (82f:10059)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 11D72, 11G05

Retrieve articles in all journals with MSC: 11D72, 11G05

Additional Information

Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society