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On the Diophantine Equations (2n − 1)(6n − 1) = x 2 and (a n − 1)(a kn − 1) = x 2

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Abstract

In this paper we prove that the equation (2n − 1)(6n − 1) = x 2 has no solutions in positive integers n and x. Furthermore, the equation (a n − 1) (a kn − 1) = x 2 in positive integers a > 1, n, k > 1 (kn > 2) and x is also considered. We show that this equation has the only solutions (a,n,k,x) = (2,3,2,21), (3,1,5,22) and (7,1,4,120).

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Authors and Affiliations

  1. Institute of Mathematics and Informatics, University of Debrecen, Debrecen, P.O. Box 12, H-4010, Hungary

    Lajos Hajdu

  2. Institute of Mathematics Sopron, Bajcsy, University of West Hungary, Zs. U. 4, H-9400, Hungary

    László Szalay

Authors
  1. Lajos Hajdu
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  2. László Szalay
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Hajdu, L., Szalay, L. On the Diophantine Equations (2n − 1)(6n − 1) = x 2 and (a n − 1)(a kn − 1) = x 2 . Periodica Mathematica Hungarica 40, 141–145 (2000). https://doi.org/10.1023/A:1010335509489

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  • DOI: https://doi.org/10.1023/A:1010335509489

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