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On an equation of Goormaghtigh. (English) Zbl 0896.11010
The equation of Goormaghtigh asks for integers that can be written with all digits 1 with respect to two distinct bases. It has been conjectured that this problem has only finitely many solutions. For fixed positive integers $m>2$ and $n>2$ in the equation $${x^m-1 \over x-1} ={y^n-1 \over y-1}, \tag 1$$ {\it H. Davenport}, {\it D. J. Lewis} and {\it A. Schinzel} proved in [J. Math., Oxf. II. Ser. 12, 304-312 (1961; Zbl 0121.28403)] that indeed only finitely many solutions in integers $x>1$ and $y>1$ with $x\ne y$ exist. They also showed that their ineffective result can be made effective by adding the condition $\text{gcd} (m-1,n-1)>1$. The present paper extends this result as follows: Theorem. Let $\text{gcd} (m-1,n-1) =d\ge 2$, and let $m-1=dr$, $n-1=ds$. Then (1) implies that $\max (x,y,m,n)$ is bounded by an effectively computable number depending only on $r$ and $s$. The proof depends on the theory of linear forms in logarithms.

11D61Exponential diophantine equations
11J86Linear forms in logarithms; Baker’s method
Full Text: EuDML