• A note on the exponential diophantine equation $(a^{n} − 1)(b^{n} − 1) = x^{2}$

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      https://www.ias.ac.in/article/fulltext/pmsc/129/05/0069

    • Keywords

       

      Pell equation; exponential diophantine equation; Lucas sequence

    • Abstract

       

      In 2002, Luca and Walsh (J. Number Theory 96 (2002) 152–173) solved the diophantine equation for all pairs $(a, b)$ such that $2\leq a$ < $b\leq 100$ with some exceptions. There are sixty nine exceptions. In this paper, we give some new results concerning the equation $(a^{n}−1)(b^{n}−1) = x^{2}$. It is also proved that this equation has no solutions if $a, b$ have opposite parity and $n$ >$4$ with $2|n$. Here, the equation is also solved for the pairs $(a, b) = (2, 50), (4, 49), (12, 45), (13, 76), (20, 77), (28, 49), (45, 100)$. Lastly, we show that when b is even, the equation $(a^{n} − 1)(b^{2n}a^{n} − 1) = x^{2}$ has no solutions $n, x$.

    • Author Affiliations

       

      REFIK KESKIN1

      1. Department of Mathematics, Faculty of Arts and Science, Sakarya University, Sakarya, Turkey
    • Dates

       
  • Proceedings – Mathematical Sciences | News

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