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 \(n\).

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 \(n\).

Reviewer: R.J.Stroeker (Rotterdam)

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\textit{T. N. Shorey}, in: Number theory and related topics. Papers presented at the Ramanujan birth centenary international colloquium, Bombay, India, 4-11 January, 1988. Oxford: Oxford University Press, published for the Tata Institute of Fundamental Research. 217--229 (1989; Zbl 0748.11022)