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On the Oesterlé-Masser conjecture. (English) Zbl 0597.10042

For any positive integers x, y and z define \(G=G(x,y,z)\) to be the product of the primes which divide xyz. Oesterlé asked if there exists a constant \(c_ 1\) such that for all positive integers x, y and z with \((x,y,z)=1\) and \(x\geq y\geq z\) we have \(x<G^{c_ 1}\). Masser later conjectured that for any positive real number \(\epsilon\), \(x<c_ 2(\epsilon)G^{1+\epsilon}\) where \(c_ 2(\epsilon)\) is a positive number which depends on \(\epsilon\) only.
In this article we prove, with the above hypotheses, that log x\(<c_ 3 G^{15}\) where \(c_ 3\) is an effectively computable constant. Further, let \(\delta\) be a positive real number. We show that there exist infinitely many positive integers x, y and z such that \(x=y+z\), \((x,y,z)=1\) and \[ x > G\quad \exp ((4-\delta)(\log G)^{1/2}/\log \log G). \]

MSC:

11N05 Distribution of primes
11J81 Transcendence (general theory)
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References:

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