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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