Duke Mathematical Journal

On the abc conjecture, II

C. L. Stewart and Kunrui Yu

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Abstract

Let x, y, and z be coprime positive integers with x+y=z. In this paper we give upper bounds for z in terms of the greatest square-free factor of xyz.

Article information

Source
Duke Math. J. Volume 108, Number 1 (2001), 169-181.

Dates
First available in Project Euclid: 5 August 2004

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1091737127

Mathematical Reviews number (MathSciNet)
MR1831823

Digital Object Identifier
doi:10.1215/S0012-7094-01-10815-6

Zentralblatt MATH identifier
1036.11032

Subjects
Primary: 11D75: Diophantine inequalities [See also 11J25]
Secondary: 11J25: Diophantine inequalities [See also 11D75] 11J86: Linear forms in logarithms; Baker's method

Citation

Stewart, C. L.; Yu, Kunrui. On the abc conjecture, II. Duke Mathematical Journal 108 (2001), no. 1, 169--181. doi:10.1215/S0012-7094-01-10815-6. http://projecteuclid.org/euclid.dmj/1091737127.


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References

  • A. Baker, ``Logarithmic forms and the $abc$-conjecture'' in Number Theory (Eger, Hungary, 1996), de Gruyter, Berlin, 1998, 37--.
  • C. L. Stewart and K. Yu, On the $abc$ conjecture, Math. Ann. 291 (1991), 225--230. MR 92k:11037
  • P. Vojta, Diophantine Approximations and Value Distribution Theory, Lecture Notes in Math. 1239, Springer, Berlin, 1987. MR 91k:11049
  • M. Waldschmidt, A lower bound for linear forms in logarithms, Acta Arith. 37 (1980), 257--283. MR 82h:10049
  • K. Yu, Linear forms in $p$-adic logarithms, II, Compositio Math. 74 (1990), 15--113. MR 91h:11065a
  • --. --. --. --., $p$-adic logarithmic forms and group varieties, I, J. Reine Angew. Math. 502 (1998), 29--92. MR 99g:11092
  • --. --. --. --., $p$-adic logarithmic forms and group varieties, II, Acta Arith. 89 (1999), 337--378. MR 2000e:11097

See also

  • See also: C. L. Stewart, Kunrui Yu. On the abc conjecture. Math. Ann. Vol. 291, No. 2 (1991), p. 225-230.