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Some remarks on the \(abc\)-conjecture. (English) Zbl 0804.11006
This paper has two main results, both related to the \(abc\) conjecture of Masser and Oesterlé. This conjecture asserts that for all \(\varepsilon>0\) there exists \(C (\varepsilon)>0\) such that for all relatively prime triples \((a,b,c)\) of integers with \(a + b + c = 0\), \(\max (| a |, | b |, | c |) \geq C (\varepsilon) r (abc)^{1+\varepsilon}\), where \(r(n)\) denotes the product of the primes dividing \(n\) (without multiplicity).
The paper’s first result concerns a possible generalization of the \(abc\) conjecture to sums \(a_ 1 + \cdots + a_ n = 0\) with \((a_ 1, \dots, a_ n) = 1\). In that case one would try to prove that \(\max (| a_ i |) \leq C \cdot r (a_ 1 \dots a_ n)^ L\) for some exponent \(L\); the authors construct an example showing that any such \(L\) must satisfy \(L \geq 2n-5\). They also conjecture that \(2n-5 + \varepsilon\) is the best possible value (with \(C\) depending on \(\varepsilon)\).
For the second main result of the paper, the authors describe an algorithm for finding examples of \((a,b,c)\) in the \(abc\) conjecture with \(\log \max (| a |, | b |, | c |)/ \log r (abc)\) large. The algorithm employs convergents in continued fraction expansions of \(\root n \of {k}\). A table is given.
Reviewer: P.Vojta (Berkeley)

MSC:
11A99 Elementary number theory
11D04 Linear Diophantine equations
11Y16 Number-theoretic algorithms; complexity
11A55 Continued fractions
11C08 Polynomials in number theory
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