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The abc–conjecture for Algebraic Numbers

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Abstract

The abc–conjecture for the ring of integers states that, for every ε > 0 and every triple of relatively prime nonzero integers (a, b, c) satisfying a + b = c, we have max(|a|, |b|, |c|) ≤ rad(abc)1 + ε with a finite number of exceptions. Here the radical rad(m) is the product of all distinct prime factors of m.

In the present paper we propose an abc–conjecture for the field of all algebraic numbers. It is based on the definition of the radical (in Section 1) and of the height (in Section 2) of an algebraic number. From this abc–conjecture we deduce some versions of Fermat's last theorem for the field of all algebraic numbers, and we discuss from this point of view known results on solutions of Fermat's equation in fields of small degrees over ℚ.

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References

  1. Masser, D. W.: Open problems, Proc. Symp. Analytic Number Theory, ed. by W. W. L. Chen, Imperial College, London, 1985

  2. Oesterlé, J.: Nouvelles approches du “théorème” de Fermat, Séminaire Bourbaki, Exp. No. 694. Astérisque, 4, 161–162, 165–186 (1988)

    Google Scholar 

  3. Elkies, N. D.: ABC implies Mordell. Internat. Math. Res. Notices, 7, 99–109 (1991)

    Article  MathSciNet  Google Scholar 

  4. Lang, S.: Elliptic Functions, Addison–Wesley Publ. Comp., London, 1973

  5. Evertse, J. H.: On equations in S-units and the Thue–Mahler equation. Invent. Math., 75, 561–584 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  6. Silverman, J. H.: The theory of height functions, in: Arithmetic Geometry, Ch. VI, eds. G. Cornell, J.H. Silverman, Springer, New York, 151–166, 1986

  7. Broberg, N.: Some examples related to the abc-conjecture for algebraic number fields. Math. Comp., 69(232), 1707–1710, (2000)

    Article  MATH  MathSciNet  Google Scholar 

  8. Browkin, J.: The abc-conjecture, in: Number Theory, eds. R. P. Bambah, V. C. Dumir, R. J. Hans–Gill, Hindustan Book Agency & Indian National Academy, New Delhi, 75–105, 2000, reprinted by Birkhäuser Verlag, Basel, 2000

  9. Frey, G.: Links between solutions of A – B = C and elliptic curves. Lecture Notes in Math., 1380, 31–62 (1989)

    Article  MathSciNet  Google Scholar 

  10. Granville, A., Stark, H.: ABC implies no “Siegel zeros” for L-functions of characters with negative discriminant. Invent. Math., 139, 509–523 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  11. Vojta, P.: Diophantine Approximation and Value Distribution Theory. Lecture Notes in Math., Springer-Verlag, 1239, (1987)

  12. Masser, D. W.: On abc and discriminants. Proc. Amer. Math. Soc., 130(11), 3141–3150 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Frey, G.: On ternary equations of Fermat type and relations with elliptic curves, in: Modular Forms and Fermat’s Last Theorem, (Boston, MA, 1995), eds. G. Cornell, J.H. Silverman, G. Stevens, Springer, New York, 527–548, 1977

  14. Dubickas, A.: On a height related to the abc-conjecture . Indian J. Pure and Appl. Math., 34, 853–857 (2003)

    MATH  MathSciNet  Google Scholar 

  15. Browkin, J., Filaseta, M., Greaves, G., Schinzel, A.: Squarefree values of polynomials and the abc-conjecture, in: Sieve Methods, Exponential Sums, and Their Applications to Number Theory; London Math. Soc. Lecture Note Ser. 237, eds. G.R.H. Greaves, G. Harman, M. N. Huxley, Cambridge University Press, Cambridge, 65–85, 1997

  16. Greaves, G., Nitaj, A.: Some polynomial identities related to the abc-conjecture, in: Number Theory in Progress (Zakopane–Kościelisko, 1997), eds. K. Győry, H. Iwaniec, J. Urbanowicz, de Gruyter, Berlin, 1, 229–236, 1999

  17. Filaseta, M., Konyagin, S.: On a limit point associated with the abc-conjecture . Colloq. Math., 76, 265–268 (1998)

    MATH  MathSciNet  Google Scholar 

  18. Dokchitser, T.: LLL & ABC. arXiv:math., NT/0307322 r2, 11 Dec 2003; Number Theory, 107, 161–167 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  19. Debarre, O., Klassen, M. J.: Points of low degree on smooth plane curves. Journ. Reine Angew. Math., 446, 81–87 (1994)

    MATH  MathSciNet  Google Scholar 

  20. Klassen, M., Tzermias, P.: Algebraic points of low degree on Fermat’s quintic. Acta Arith., 82, 393–401 (1997)

    MATH  MathSciNet  Google Scholar 

  21. Hao, F. H., Parry, C. J.: The Fermat equation over quadratic fields. J. Number Theory, 19, 115–130 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  22. Hilbert, D.: Die Theorie der algebraischen Zahlkörper. Jahresber. Deutsch. Math. -Verein, 4, 175–546 (1897)

    Google Scholar 

  23. Kolyvagin, V. A.: Fermat’s equations over cyclotomic fields. Trudy Inst. Mat. V. A. Steklova, 208, 163–185 (1995) (Russian)

    MathSciNet  Google Scholar 

  24. Aigner, A.: Über die Möglichkeit von x 4 + y 4 = z 4 in quadratischen Körpern. Jahresber. Deutsch. Math.-Verein, 43, 226–229 (1934)

    MATH  Google Scholar 

  25. Aigner, A.: Die Unmöglichkeit von x 6 + y 6 = z 6 und x 9 + y 9 = z 9 in quadratischen Körpern. Monatsh. Math., 61, 147–150 (1957)

    Article  MATH  MathSciNet  Google Scholar 

  26. Gross, B. H., Rohrlich, D. E.: Some results on the Mordell–Weil group of the jacobian of the Fermat’s curve. Invent. Math., 44, 201–224 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  27. Tzermias, P.: Algebraic points of low degree on the Fermat curve of degree seven. Manuscripta Math., 97(4), 483–488 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  28. Sall, O.: Points algébriques de petit degré sur les courbes de Fermat. C. R. Acad. Sci. Paris, Série I, 330, 67–70 (2000)

    MATH  MathSciNet  Google Scholar 

  29. Ribenboim, P.: Fermat’s Last Theorem for Amateurs, Springer-Verlag, New York, 1999

  30. Terjanian, G.: Sur la loi de réciprocité des puissances l-èmes. Acta Arith., 54(2), 87–125 (1989)

    MATH  MathSciNet  Google Scholar 

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  1. Institute of Mathematics, University of Warsaw, ul. Banacha 2, PL–02–097, Warsaw, Poland

    Jerzy Browkin

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  1. Jerzy Browkin
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Browkin, J. The abc–conjecture for Algebraic Numbers. Acta Math Sinica 22, 211–222 (2006). https://doi.org/10.1007/s10114-005-0624-3

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