On the solutions of some Lebesgue-Ramanujan-Nagell type equations

Denote by h = h(-p) the class number of the imaginary quadratic field Q(root-p) with p prime. It is well known that h = 1 for p is an element of{3, 7, 11, 19, 43, 67, 163}. Recently, all the solutions of the Diophantine equation x(2) + p(infinity) = 4yn with h = 1 were given by Chakraborty et al. in [Complete solutions of certain Lebesgue-Ramanujan-Nagell type equations, Publ. Math. Debrecen 97(3-4) (2020) 339-352]. In this paper, we study the Diophantine equation x(2) + p(infinity) = 2ryn in unknown integers (x,y,s,r,n), where s >= 0, r >= 3, n >= 3, h is an element of{1, 2, 3} and gcd(x,y) = 1. To do this, we use the known results from the modularity of Galois representations associated with Frey-Hellegoaurch elliptic curves, the symplectic method and elementary methods of classical algebraic number theory. The aim of this paper is to extend the above results of Chakraborty et al.