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On the diophantine equation $(x^3-1)/(x-1)=(y^n-1)/(y-1)$


Author: Maohua Le
Journal: Trans. Amer. Math. Soc. 351 (1999), 1063-1074
MSC (1991): Primary 11D61, 11J86
DOI: https://doi.org/10.1090/S0002-9947-99-02013-9
MathSciNet review: 1443198
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we prove that the equation $(x^3-1)/(x-1)=$ $(y^n-1)/(y-1)$, $x,y,n\in \mathbb {N}$, $x>1$, $y>1$, $n>3$, has only the solutions $(x,y,n)=(5,2,5)$ and $(90,2,13)$ with $y$ is a prime power. The proof depends on some new results concerning the upper bounds for the number of solutions of the generalized Ramanujan-Nagell equations.


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  • Roger ApĂ©ry, Sur une Ă©quation diophantienne, C. R. Acad. Sci. Paris 251 (1960), 1263–1264 (French). MR 120194
  • Roger ApĂ©ry, Sur une Ă©quation diophantienne, C. R. Acad. Sci. Paris 251 (1960), 1451–1452 (French). MR 120193
  • V. I. Baulin, On an indeterminate equation of the third degree with least positive discriminant, Tulâ€Čsk. Gos. Ped. Inst. Učen. Zap. Fiz.-Mat. Nauk Vyp. 7 (1960), 138–170 (Russian). MR 0199149
  • R. Goormaghtigh, L’intermĂ©diaire des MathĂ©maticiens, 24 (1917), 88.
  • Richard K. Guy, Unsolved problems in number theory, Unsolved Problems in Intuitive Mathematics, vol. 1, Springer-Verlag, New York-Berlin, 1981. Problem Books in Mathematics. MR 656313
  • Michel Laurent, Maurice Mignotte, and Yuri Nesterenko, Formes linĂ©aires en deux logarithmes et dĂ©terminants d’interpolation, J. Number Theory 55 (1995), no. 2, 285–321 (French, with English summary). MR 1366574, DOI https://doi.org/10.1006/jnth.1995.1141
  • M.-H. Le, The divisibility of the class number for a class of imaginary quadratic fields, Kexue Tongbao, 32 (1987), 724–727. (in Chinese)
  • M.-H. Le, The diophantine equation $D_1x^2+D_2=2^{n+2}$, Acta Arith., 64 (1993), 29–41.
  • Mao Hua Le, A note on the generalized Ramanujan-Nagell equation, J. Number Theory 50 (1995), no. 2, 193–201. MR 1316814, DOI https://doi.org/10.1006/jnth.1995.1013
  • Rudolf Lidl and Harald Niederreiter, Finite fields, Encyclopedia of Mathematics and its Applications, vol. 20, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1983. With a foreword by P. M. Cohn. MR 746963
  • T. Nagell, Contributions to the theory of a category of diophantine equations of the second degree with two unknowns, Nova Acta Soc. Sci. Upsal., (4) 16, no. 2, 38pp. (1955).
  • T. Nagell, The diophantine equation $x^2+7=2^n$, Arkiv. Mat., 4 (1960), 185–187.
  • R. Ratat, L’Intermediaire des MathĂ©maticiens, 23 (1916), 150.
  • T. N. Shorey, Some exponential Diophantine equations. II, Number theory and related topics (Bombay, 1988) Tata Inst. Fund. Res. Stud. Math., vol. 12, Tata Inst. Fund. Res., Bombay, 1989, pp. 217–229. MR 1441334

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Additional Information

Maohua Le
Affiliation: Department of Mathematics, Zhanjiang Teachers College, Postal Code 524048, Zhanjiang, Guangdong, P. R. China

Additional Notes: Supported by the National Natural Science Foundation of China and the Guangdong Provincial Natural Science Foundation
Article copyright: © Copyright 1999 American Mathematical Society