A Note On the Exponential Diophantine Equation (a^n-1)(b^n-1)=x^2 (1801.04717v1)

Abstract: In 2002, F. Luca and G. Walsh solved the Diophantine equation in the title for all pairs (a,b) such that 1<a<b\<101 with some exceptions. There are sixty nine exceptions. In this paper, we give some new results concerning the equation in the title. It is proved that the equation (a^n-1)(b^n-1)=x^2 has no solutions if a,b have opposite parity and n\>4 with 2|n. Also, we solved (a^n-1)(b^n-1)=x^2 for the pairs (a,b)=(2,50),(4,49),(12,45),(13,76),(20,77),(28,49), and (45,100). Lastly, we show that when b is even, the equation (a^n-1)(b^2na^n-1)=x^2 has no solutions n,x.