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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
Full Text:
References:
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