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 and 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 and with exist. They also showed that their ineffective result can be made effective by adding the condition .
The present paper extends this result as follows: Theorem. Let , and let , . Then (1) implies that is bounded by an effectively computable number depending only on and . The proof depends on the theory of linear forms in logarithms.