A note on the number of solutions of the generalized Ramanujan-Nagell equation \(D_ 1x^ 2+D_ 2=4p^ n\)
Let \(D_1\), \(D_2\) be positive integers with \(2\nmid D_1D_2\) and \(\text{gcd} (D_1, D_2)=1\). Let \(p\) be a prime with \(p\nmid D_1D_2\). It is proved that the generalized Ramanujan-Nagell equation of the title has at most two positive integer solutions \((x,n)\) except for the cases \((D_1,D_2,p) =(1,7,2)\), \((3,5,2)\), \((1,11,3)\) and \((1,19,5)\).
scientific article; zbMATH DE number 992095