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Some exponential diophantine equations. II. (English) Zbl 0748.11022
Number theory and related topics, Pap. Ramanujan Colloq., Bombay/India 1988, Stud. Math., Tata Inst. Fundam. Res. 12, 217-229 (1989).
[For the entire collection see Zbl 0743.00038. For Part I, cf. New advances in transcendence theory, Proc. Symp., Durham/UK 1986, 352-365 (1988; Zbl 0658.10024).]
This paper consists of three independent parts. First, let \(s(N)\) be the number of integers \(y\), \(1<y<N-1\) such that \(N\) has all its digits equal to 1 with respect to the base \(y\). The author reviews what is known about the possible values \(s(N)\) can take and some new results are obtained.
The second part starts with a brief discussion of the occurrence of perfect powers in binary recurrences. Also it is indicated how it can be shown that \(\tau(p^ m)=\tau(p^ n)\) with \(m\neq n\) implies that \(\max(m,n,p)\) is bounded from above by an effectively computable absolute constant (here \(p\) is prime and \(\tau(p)\neq 0\), where \(\tau\) is Ramanujan’s \(\tau\)-function).
In the final part the author deals with the exponential equation \[ m(m+d)\ldots(m+(k-1)d)=by^ \ell, (*) \] for positive integers \(b\), \(d\), \(m\), \(y\), \(k>2\), \(\ell\geq 2\) with \((m,d)=1\) and \(P(b)\leq k\). A complete proof is given of the following result: there exists an effectively computable absolute constant \(c_{18}>0\) such that (*) with \(\ell\geq c_{18}\) implies that \(k\) is bounded by an effectively computable number depending only on \(m\) and \(n\).

MSC:
11D61 Exponential Diophantine equations
11B37 Recurrences
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