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Perfect powers in products of terms in an arithmetical progression. (English) Zbl 0708.11021
The authors consider the diophantine equation \[ (*)\quad m(m+d)... (m+(k-1)d)=by^{\ell} \] in positive integers b, d, k, \(\ell\), m, y, subject to the conditions: P(b)\(\leq k\), \(\gcd (m,d)=1\), \(k>2\), \(\ell >1\), \(y>1\) and \(P(y)>k\). Further, it is assumed that \(\ell\) is prime. (N.B. P(x) denotes the greatest prime factor of x.)
A comprehensive overview is given of the present state of knowledge concerning equation (*). Considerable improvements are obtained of recent results of R. Marszalek [Monatsh. Math. 100, 215-222 (1985; Zbl 0582.10011)] and the first author [New advances in transcendence theory, Proc. Symp. Durham 1986, 352-365 (1988; Zbl 0658.10024)]. A sample result, incorporated in the authors’ “main aim of this paper”, is the following:
Let equation (*) be satisfied. If \(\ell \in \{2,3,5\}\) then k is bounded by an effectively computable number depending only on \(\omega\) (d). If \(\ell \geq 7\) then k is bounded by an effectively computable number depending only on \(\ell\) and \(\omega (d_ 1)\), where \(d_ 1\) is the maximal divisor of d such that all prime factors of \(d_ 1\) are \(\equiv 1(mod \ell).\)
The proofs use a result of J.-H. Evertse [Compos. Math. 47, 289-315 (1982; Zbl 0498.10014)] when \(\ell =3\) and \(\ell =5\). Otherwise, the proofs are elementary.
Reviewer: R.J.Stroeker

MSC:
11D61 Exponential Diophantine equations
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