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

Abstract

Denote by $h=h(-p)$ the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-p})$ with $p$ prime. It is well known that $h=1$ for $p\in \{3,7,11,19,43,67,163\}$. Recently, all the solutions of the Diophantine equation $x^2+p^s=4y^n$ with $h=1$ were given by Chakraborty et al. in [Complete solutions of certain Lebesgue–Ramanujan–Nagell type equations, Publ. Math. Debrecen97(3–4) (2020) 339–352]. In this paper, we study the Diophantine equation $x^2+p^s=2^r{y}^n$ in unknown integers $(x,y,s,r,n),$ where $s\ge 0$, $r\ge 3$, $n\ge 3$, $h\in \{1,2,3\}$ 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.