Ideal 9th-order multigrades and Letac’s elliptic curve
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- by C. J. Smyth PDF
- Math. Comp. 57 (1991), 817-823 Request permission
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
- J. W. S. Cassels, Diophantine equations with special reference to elliptic curves, J. London Math. Soc. 41 (1966), 193–291. MR 199150, DOI 10.1112/jlms/s1-41.1.193
- Noam D. Elkies, On $A^4+B^4+C^4=D^4$, Math. Comp. 51 (1988), no. 184, 825–835. MR 930224, DOI 10.1090/S0025-5718-1988-0930224-9
- William Fulton, Algebraic curves. An introduction to algebraic geometry, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York-Amsterdam, 1969. Notes written with the collaboration of Richard Weiss. MR 0313252
- Albert Gloden, Mehrgradige Gleichungen, P. Noordhoff, Groningen, 1944 (German). 2d ed; Mit einem Vorwort von Maurice Kraitchik. MR 0019638 T. Nagell, Solution de quelque problèmes dans la théorie arithmétique des cubiques planes du premier genre, Vid. Akad. Skrifter Oslo 1 (1935), no. 1.
- Elmer Rees and Christopher Smyth, On the constant in the Tarry-Escott problem, Cinquante ans de polynômes (Paris, 1988) Lecture Notes in Math., vol. 1415, Springer, Berlin, 1990, pp. 196–208. MR 1044114, DOI 10.1007/BFb0084888
- Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210, DOI 10.1007/978-1-4757-1920-8 E. M. Wright, personal communication, 1989.
- E. M. Wright, The Tarry-Escott and the “easier” Waring problems, J. Reine Angew. Math. 311(312) (1979), 170–173. MR 549963, DOI 10.1515/crll.1979.311-312.170
Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Math. Comp. 57 (1991), 817-823
- MSC: Primary 11D72; Secondary 11G05
- DOI: https://doi.org/10.1090/S0025-5718-1991-1094960-9
- MathSciNet review: 1094960