A note on the exponential diophantine equation (an-1)(bn-1)=x2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{(a^{n}-1)(b^{n}-1)=x^{2}}$$\end{document}

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} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \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} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \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} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \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} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \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} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ (a^{n}-1)(b^{2n}a^{n}-1)=x^{2}$$\end{document} has no solutions n, x.