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

• Refik Keskin
Article

## 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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