Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0896.11010
Nesterenko, Yu.V.; Shorey, T.N.
On an equation of Goormaghtigh. (English)
[J] Acta Arith. 83, No.4, 381-389 (1998). ISSN 0065-1036; ISSN 1730-6264/e

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 $\text {gcd} (m-1,n-1)>1$.\par 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.
[R.J.Stroeker (Rotterdam)]

MSC 2000:: *11D61 Exponential diophantine equations
11J86 Linear forms in logarithms; Baker's method

Keywords: higher degree diophantine equations; equation of Goormaghtigh; linear forms in logarithms

Citations: Zbl 0121.28403

Cited in: Zbl 1047.11028

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]