*(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

*H. Davenport*, *D. J. Lewis* and *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.