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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 x-1=y n -1 y-1,(1)

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 xy exist. They also showed that their ineffective result can be made effective by adding the condition gcd(m-1,n-1)>1.

The present paper extends this result as follows: Theorem. Let gcd(m-1,n-1)=d2, 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.


MSC:
11D61Exponential diophantine equations
11J86Linear forms in logarithms; Baker’s method