On the Diophantine equation \(\frac{x^m-1}{x-1}=\frac{y^n-1}{y-1}\).
The main result of this paper is: for fixed rational integers \(y>x>1\), the exponential Diophantine equation \({x^m-1\over x-1}={y^n-1\over y-1}\) has at most two solutions (of course \(m\) and \(n\) are \({}>1\)). Moreover, the authors show that it has also finitely many solutions \((x,y,m,n)\) with \(m>2\) and \(n>2\) for which \(\gcd(m-1,n-1)>1\) and the quotient \((m-1)/(n-1)\) is bounded. This equation was considered in 1917 by \textit{J. Rose} and \textit{R. Goormaghtigh} [Interméd. des math. 24, 88--90 (1917; JFM 46.0230.13)], who noticed the two solutions \((x,y,m,n)=(2,5,5,3)\) and \((2,90,13,3)\). Results on this equation were obtained in particular by \textit{H. Davenport, D. J. Lewis} and \textit{A. Schinzel} [Q. J. Math., Oxf. II. Ser. 12, 304--312 (1961; Zbl 0121.28403)] and \textit{Yu. V. Nesterenko} and \textit{T. N. Shorey} [Acta Arith. 83, 381--389 (1998; Zbl 0896.11010)]. The methods of the proofs of the present paper combine Baker's theory of linear forms in logarithms (both in the complex and \(p\)-adic cases) and use of irrationality measures of certain roots of rational numbers (due to A. Baker).
number of solutions of Diophantine equations
scientific article; zbMATH DE number 2101013