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

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