In 2002, Luca and Walsh (J. Number Theory96
(2002) 152–173) solved the diophantine equation for all pairs (a, b) such that 2≤a<b≤100\documentclass[12pt]{minimal}
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\begin{document}$$2\le a<b\le 100$$\end{document} with some exceptions. There are sixty nine exceptions. In this paper, we give some new results concerning the equation (an-1)(bn-1)=x2\documentclass[12pt]{minimal}
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\begin{document}$$(a^{n}-1)(b^{n}-1)=x^{2}$$\end{document}. It is also proved that this equation has no solutions if a, b have opposite parity and n>4\documentclass[12pt]{minimal}
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\begin{document}$$n>4$$\end{document} 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)\documentclass[12pt]{minimal}
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\begin{document}$$(a,b)=(2,50),(4,49),(12,45),(13,76),(20,77),(28,49),(45,100)$$\end{document}. Lastly, we show that when b is even, the equation (an-1)(b2nan-1)=x2\documentclass[12pt]{minimal}
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\begin{document}$$ (a^{n}-1)(b^{2n}a^{n}-1)=x^{2}$$\end{document} has no solutions n, x.
