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Some exponential diophantine equations. (English) Zbl 0658.10024
New advances in transcendence theory, Proc. Symp., Durham/UK 1986, 352-365 (1988).
[For the entire collection see Zbl 0644.00005.]
In this paper an overview is given of recent results on certain exponential diophantine equations, obtained by the author by combining the theory of linear forms in logarithms and Baker’s effective irrationality measures. These results are published elsewhere [Perfect powers in values of certain polynomials at integer points, Math. Proc. Camb. Philos. Soc. 99, 195-207 (1986; Zbl 0598.10029); On the equation \(z^ q=(x^ n-1)/(x-1)\), Indagationes Math. 48, 345-351 (1986; Zbl 0603.10018); On the equation \(ax^ m-by^ n=k\), ibid., 48, 353-358 (1986; Zbl 0603.10019); Perfect powers in products of integers from a block of consecutive integers, Acta Arith. 49, 71-79 (1987; Zbl 0582.10012)].
Also the following generalization of the Erdős-Selfridge equation is studied, namely \[ (*)\quad (m+d)... (m+kd)=by^{\ell},\quad (m,d)=1. \] A detailed proof is given of the following Theorem. If (*) is satisfied with positive integers b,d,m,y,k\(\geq 2\), and \(\ell \geq 3\), then k is bounded by an effectively computable absolute constant if at least one of the following conditions is satisfied: \((1)\quad m+d>k\) and \((\ell,\prod_{p| d}(p-1))=1,\) \((2)\quad m+d>k\) and the largest prime divisor of d is fixed, \((3)\quad m+d\leq k\) and \(b=1\).
Reviewer: R.J.Stroeker

MSC:
11D61 Exponential Diophantine equations
11-02 Research exposition (monographs, survey articles) pertaining to number theory