Some exponential Diophantine equations III: a new look at the generalized Lebesgue–Nagell equation

Abstract

Let D be a fixed non-square integer, and let h(4D) denote the class number of binary quadratic primitive forms with discriminant 4D. Let k be a fixed even integer with $\gcd (D,k)=1$. In this paper, using some properties on exponential Diophantine equations with the forms $X^2-DY^2=k^Z$ and ${X^\prime }^2-D{Y^\prime }^2=4k^{Z^\prime }$, we prove that if the equation $a^2-Db^2=8\zeta$ has no integer solutions (a, b) with $\gcd (a,b)=1$, where $\zeta =1$ or 2 according to $2\not \mid h(4D)$ or $2\mid h(4D)$, then the generalized Lebesgue–Nagell equation $(*)$ $x^2-D^m=y^n$ has no positive integer solutions (x, y, m, n) with $\gcd (x,y)=1$, $2\mid y$, $2\not \mid m$, $n>2$ and $h(4D)\mid n$. By the above result, we can directly derive that if $D<0$ and $D\ne -7$ or $-15$, then $(*)$ has no positive integer solutions (x, y, m, n) with $\gcd (x,y)=1$, $2\mid y$, $2\not \mid m$, $n>2$ and $h(4D)\mid n$.