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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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