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 X2-DY2=kZ and X′2-DY′2=4kZ′, we prove that if the equation a2-Db2=8ζ has no integer solutions (a, b) with gcd(a,b)=1, where ζ=1 or 2 according to 2∤h(4D) or 2∣h(4D), then the generalized Lebesgue–Nagell equation (∗)x2-Dm=yn has no positive integer solutions (x, y, m, n) with gcd(x,y)=1, 2∣y, 2∤m, n>2 and h(4D)∣n. By the above result, we can directly derive that if D<0 and D≠-7 or -15, then (∗) has no positive integer solutions (x, y, m, n) with gcd(x,y)=1, 2∣y, 2∤m, n>2 and h(4D)∣n.