Abstract
The Diophantine equation \(\frac{x^n-1}{x-1}=y^q\) has four known solutions in integers \(x, y, q\) and \(n\) with \(|x|, |y|, q > 1\) and \(n > 2\). Whilst we expect that there are, in fact, no more solutions, such a result is well beyond current technology. In this paper, we prove that if \((x,y,n,q)\) is a solution to this equation, then \(n\) has three or fewer prime divisors, counted with multiplicity. This improves a result of Bugeaud and Mihăilescu.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bennett, M.A.: Rational approximation to algebraic numbers of small height: the Diophantine equation \(\vert ax^n-by^n\vert =1\). J. Reine Angew. Math. 535, 1–49 (2001)
Bugeaud, Y., Hanrot, G., Mignotte, M.: Sur l’équation diophantienne \(\frac{x^n-1}{x-1}=y^q\). III. Proc. London Math. Soc. 84, 59–78 (2002)
Bugeaud, Y., Mignotte, M.: Sur l’équation diophantienne \(\frac{x^n-1}{x-1}=y^q\), II. C. R. Acad. Sci. Paris Sr. I Math. 328, 741–744 (1999)
Bugeaud, Y., Mignotte, M.: L’équation de Nagell–Ljunggren \(\frac{x^n-1}{x-1} = y^q\). Enseign. Math. 48, 147–168 (2002)
Bugeaud, Y., Mignotte, M., On the Diophantine equation \((x^n-1), (x-1) = y^q\) with negative \(x\). Number theory for the millennium, I (Urbana, IL, : 145–151. A K Peters, Natick, MA, 2000) (2002)
Bugeaud, Y., Mignotte, M., Roy, Y.: On the Diophantine equation \((x^n-1)/(x-1)=y^q\). Pac. J. Math. 193, 257–268 (2000)
Bugeaud, Y., Mihăilescu, P.: On the Nagell–Ljunggren equation \(\frac{x^n-1}{x-1}=y^q\). Math. Scand. 101, 177–183 (2007)
Guralnick, R.M.: Subgroups of prime power index in a simple group. J. Algebra 81, 304–311 (1983)
Ljunggren, W.: Noen Setninger om ubestemte likninger av formen \((x^n-1)/(x-1)=y^q\). Norsk. Mat. Tidsskr. 25, 17–20 (1943)
Mihăilescu, P.: New bounds and conditions for the equation of Nagell–Ljunggren. J. Number Theory 124(2), 380–395 (2007)
Mihăilescu, P.: Primary cyclotomic units and a proof of Catalan’s conjecture. J. Reine Angew. Math. 572, 167–195 (2004)
Nagell, T.: Des équations indéterminées \(x^2+x+1=y^n\) et \(x^2+x+1=3y^n\). Nordsk. Mat. Forenings Skr. 2 (1920)
Nagell, T.: Note sur l’équation indéterminée \((x^n-1)/(x-1)=y^q\). Norsk. Mat. Tidsskr. 2, 75–78 (1920)
Sander, J.W.: Irrationality criteria for Mahler’s numbers. J. Number Theory 52, 145–156 (1995)
Shorey, T.N.: Perfect powers in values of certain polynomials at integer points. Math. Proc. Camb. Philos. Soc. 99, 195–207 (1986)
Acknowledgments
The authors were supported by grants from, respectively, the Natural Sciences and Engineering Council of Canada, and the National Science Foundation, Grant DMS-1102563.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by U. Zannier.
Rights and permissions
About this article
Cite this article
Bennett, M.A., Levin, A. The Nagell–Ljunggren equation via Runge’s method. Monatsh Math 177, 15–31 (2015). https://doi.org/10.1007/s00605-015-0748-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-015-0748-1