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On the $$abc$$ conjecture. (English) Zbl 0761.11030
For an integer $$n$$, define $$G(n)$$ to be the product of the primes which divide $$n$$. Then the $$abc$$ conjecture of Masser-Oesterlé asserts that for all $$\varepsilon>0$$ there is a real $$c_ 1>0$$ such that for all relatively prime $$a,b,c\in\mathbb{Z}$$ with $$a+b=c$$, $$\max(| a|,| b|,| c|)\leq c_ 1 G(abc)^{1+\varepsilon}$$. This is a quite strong conjecture which implies, among other things, the asymptotic Fermat conjecture.
This paper proves that if $$G=G(abc)$$ and $$| c|>2$$, then there exists an absolute constant $$c_ 2$$ such that $\log\max(| a|,| b|,| c|)<G^{2/3+c_ 2/\log\log G}.$ This is a strengthening of the inequality $$\log\max(| a|,| b|,| c|)<c_ 3 G^{15}$$ proved by C. L. Stewart and R. Tijdeman [Monatsh. Math. 102, 251-257 (1986; Zbl 0597.10042)]. However, it is still not strong enough for applications.
The proof is based on Baker’s theory on linear forms in logarithms; therefore the constant $$c_ 2$$ can be explicitly computed.
Reviewer: P.Vojta (Berkeley)

##### MSC:
 11J25 Diophantine inequalities 11J86 Linear forms in logarithms; Baker’s method
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##### References:
 [1] Baker, A., Stark, H.M.: On a fundamental inequality in number theory. Ann. Math. 94, 190-199 (1971) · Zbl 0219.12009 · doi:10.2307/1970742 [2] Frey, G.: Elliptic curves and solutions ofA?B=C. In: Goldstein, C. (ed.) S?minaire de Th?orie des Nombres Paris 1985-86. (Prog. Math., vol. 71, pp. 39-51) Boston Basel Stuttgart: Birkh?user 1987 [3] Mason, R.C.: Diophantine equations over function fields. (Lond. Math. Soc. Lect. Note Ser., vol. 96). Cambridge: Cambridge University Press 1984 · Zbl 0533.10012 [4] Masser, D.W.: Open problems. In: Chen, W.W.L. (ed.) Proc. Symp. Analytic Number Theory. London: Imperial College 1985 [5] Oesterl?, J.: Nouvelles approches du ?Th?or?me? de Fermat: S?minaire Bourbaki, 1987-88, no. 694. (Ast?risque, vols. 161-162, pp. 165-186). Paris: Soc. Math. Fr. 1988 [6] Van der Poorten, A.J.: Linear forms in logarithms in thep-adic case. In: Baker, A., Masser, D.W. (ed.) Transcendence Theory: Advances and Applications, pp. 29-57. London: Academic Press 1977 · Zbl 0367.10034 [7] Stewart, C.L., Tijdeman, R.: On the Oesterl?-Masser conjecture. Monatsh. Math.102, 251-257 (1986) · Zbl 0597.10042 · doi:10.1007/BF01294603 [8] Szpiro, L.: La conjecture de Mordell [d’apr?s Faltings]. S?minaire Bourbaki, 1983-84, no. 619. (Ast?risque, vols. 121-122, pp. 93-103) Paris: Soc. Math. Fr. 1985 [9] Vojta, P.: Diophantine approximations and value distribution theory. (Lect. Notes Math., vol. 1239) Berlin Heidelberg New York: Springer 1987 · Zbl 0609.14011 [10] Waldschmidt, M.: A lower bound for linear forms in logarithms. Acta Arith.37, 257-283 (1980) · Zbl 0357.10017 [11] Yu, Kunrui: Linear forms inp-adic logarithms. Acta Arith.53, 107-186 (1989) · Zbl 0699.10050 [12] Yu, Kunrui: Linear forms inp-adic logarithms. II. Compos. Math.74, 15-113 (1990) · Zbl 0723.11034
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