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On a limit point associated with the \(abc\)-conjecture. (English) Zbl 0904.11010

The \(abc\)-conjecture can be rephrased to state that when \(a+b=c\) and the positive integers \(a\), \(b\) are coprime then the double sequence \((L_{a,b})\) given by \[ L_{a,b}=\log c /\log\prod_{p\mid abc}p \] has greatest limit point 1. There is a thriving industry which makes inferences from this conjecture; in the opposite direction, the authors contribute to what is known about \((L_{a,b})\). The example \(a=1\), \(b=2^n\) shows that \((L_{a,b})\) has a (possibly infinite) limit point \(L\geq 1\): the authors show that such a limit point exists with \(1 \leq L < {3\over 2}\).
From the other side, it is known that the limit points of \((L_{a,b})\) fill the interval \(\big[{1\over 3}, {15\over 16}\big]\), also that on the \(abc\)-conjecture they would consist precisely of the points in \(\big[{1\over 3},1 \big]\). This was shown by J. Browkin, M. Filaseta, G. Greaves and A. Schinzel [in Sieve methods, exponential sums and their applications in number theory (G. R. H. Greaves, G. Harman and M. N. Huxley, eds.), London Math. Soc. Lect. Note Ser. 237, 65-85 (1997)]. Recently the entry \(15\over 16\) has been improved to \(36\over 37\) by the reviewer and A. Nitaj, in a paper to appear in [Number theory in progress: Proc. int. conf. on number theory, Zakopane, Poland, June 30–July 9 (1997)].
The authors’ method allows them to shift their interval to \((3/(3+\varepsilon), 3/(2+\varepsilon))\) for any \(\varepsilon\) in \((0,1)\), so that (for example) there must be a limit point \(L\) in \(( {36\over 37}, {147\over 101})\), although no example of such a number is currently known.

MSC:

11D75 Diophantine inequalities
11D04 Linear Diophantine equations
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