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A note on the exponential diophantine equation \(\varvec{(a^{n}-1)(b^{n}-1)=x^{2}}\)

  • Refik KeskinEmail author
Article
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Abstract

In 2002, Luca and Walsh (J. Number Theory 96 (2002) 152–173) solved the diophantine equation for all pairs (ab) such that \(2\le a<b\le 100\) 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 ab 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 nx.

Keywords

Pell equation exponential diophantine equation Lucas sequence 

Mathematics Subject Classification

11D61 11D31 11B39 

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Copyright information

© Indian Academy of Sciences 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Arts and ScienceSakarya UniversitySakaryaTurkey

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