On Diophantine equations of the form $(xⁿ -1)(y^m -1)=z²

Year, pages: 2000, 87 - 89

At the 13th Czech and Slovak Conference in Number Theory, L. Szalay showed that the Diophantine equation $(2ⁿ-1)(3ⁿ-1)=x²$ has no solutions in positive integers $n$ and $x$. Szalay's proof used the evaluation of certain Jacobi symbols to arrive at the result. The purpose of this paper is to generalize the result of Szalay by proving that the Diophantine equation $(2ⁿ-1)(3^m-1)=x²$ has no solutions in positive integers $n,m,x$. We further discuss the solvability of the more general equation $(xⁿ-1)(y^m-1)=z²$.

Walsh, P. 2000. On Diophantine equations of the form $(xⁿ -1)(y^m -1)=z². In Tatra Mountains Mathematical Publications, vol. 20, no.3, pp. 87-89. 1210-3195.

Walsh, P. (2000). On Diophantine equations of the form $(xⁿ -1)(y^m -1)=z². Tatra Mountains Mathematical Publications, 20(3), 87-89. 1210-3195.