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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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