×

On the \(abc\) conjecture. II. (English) Zbl 1036.11032

The \(abc\) conjecture of Masser and Oesterlé asserts that for all \(\varepsilon>0\) there is a constant \(C\) such that for all nonzero \(a,b,c\in\mathbb Z\) with \(a+b+c=0\) and \((a,b,c)=1\), the inequality \(H(a,b,c)\leq CG^{1+\varepsilon}\) holds, where \(H(a,b,c)=\max\{| a| ,| b| ,| c| \}\) and \(G=G(abc)\) is the product of all primes dividing \(abc\) (without multiplicity).
This paper is a sequel to the earlier paper by the authors [C. L. Stewart and K. Yu, Math. Ann. 291, 225–230 (1991; Zbl 0761.11030)]. It sharpens the bound of that paper, obtaining a bound \(H(a,b,c)<\exp(CG^{1/3}(\log G)^3)\), for an effectively computable constant \(C\).
Moreover, for \(n\in\mathbb Z\setminus\{-1,0,1\}\) let \(P(n)\) be the largest prime factor of \(n\), and also let \(P(\pm1)=1\). This paper also shows that there is an effectively computable constant \(C\) such that for all nonzero \(a,b,c\in\mathbb Z\) with \(a+b+c=0\), \((a,b,c)=1\), and \(H(a,b,c)>2\), the inequality \(H(a,b,c)<\exp(P'G^{C\log\log\log G^{*}/\log\log G})\) holds, where \(G^{*}=\max\{G,16\}\) and \(P'=\max\{P(a),P(b),P(c)\}\).
As was the case with the authors’ earlier paper, the theorems of this paper are proved using A. Baker’s theory of logarithmic forms, especially a recent estimate of K. Yu [Acta Arith. 89, 337–378 (1999; Zbl 0928.11031)].

MSC:

11J25 Diophantine inequalities
11D75 Diophantine inequalities
11J86 Linear forms in logarithms; Baker’s method
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. Baker, “Logarithmic forms and the \(abc\)-conjecture” in Number Theory (Eger, Hungary, 1996) , de Gruyter, Berlin, 1998, 37–. · Zbl 0973.11047
[2] C. L. Stewart and K. Yu, On the \(abc\) conjecture , Math. Ann. 291 (1991), 225–230. MR 92k:11037 · Zbl 0761.11030
[3] P. Vojta, Diophantine Approximations and Value Distribution Theory , Lecture Notes in Math. 1239 , Springer, Berlin, 1987. MR 91k:11049 · Zbl 0609.14011
[4] M. Waldschmidt, A lower bound for linear forms in logarithms , Acta Arith. 37 (1980), 257–283. MR 82h:10049 · Zbl 0357.10017
[5] K. Yu, Linear forms in \(p\)-adic logarithms, II , Compositio Math. 74 (1990), 15–113. MR 91h:11065a · Zbl 0723.11034
[6] –. –. –. –., \(p\)-adic logarithmic forms and group varieties, I , J. Reine Angew. Math. 502 (1998), 29–92. MR 99g:11092 · Zbl 0912.11029
[7] –. –. –. –., \(p\)-adic logarithmic forms and group varieties, II , Acta Arith. 89 (1999), 337–378. MR 2000e:11097 · Zbl 0928.11031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.