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Perfect powers in products of terms in an arithmetical progression. II. (English) Zbl 0763.11015
The authors consider the diophantine equation \(m(m+d)\cdots(m+(k- 1)d)=by^ \ell\) in positive integers \(b\), \(d\), \(k\), \(\ell\), \(m\), \(y\) subject to \(P(b)\leq k\), \(\text{gcd}(m,d)=1\), \(k>2\), \(\ell>1\) and \(P(y)>k\). As usual, \(P(x)\) denotes the greatest prime factor of \(x\). Let \(\ell\) be prime and let \(d_ 1\) denote the largest divisor of \(d\) with the property that all its prime factors are \(\equiv 1\pmod \ell\).
This paper is mainly concerned with improvements of effective results and their consequences, results on the boundedness of \(k\) in terms of \(d_ 1\) and \(\ell\), that were previously obtained by the authors [see the first author, New advances in transcendence theory, Durham 1986, 352-365 (1988; Zbl 0658.10024); the authors, Compos. Math. 76, 307-344 (1990; Zbl 0708.11021); Compos. Math. 82, 107-117 (1992; reviewed above)]. The actual results are rather technical, so that we refrain from reproducing them here.
[Part III of this series has been published in Acta Arith. 61, No. 4, 391-398 (1992)].

MSC:
11D61 Exponential Diophantine equations
11B25 Arithmetic progressions
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References:
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