On the solutions of some Lebesgue-Ramanujan-Nagell type equations (Q6554581)

In the paper, the authors consider a class of polynomial-exponential Diophantine equations of the form \(x^2 + p^s = 2^r y^n\) in integers \((x, y, s, r, n)\), where \(s \geq 0\), \(r \geq 3\), \(n \geq 3\), \(h \in \{1, 2, 3\}\), and \(\gcd(x, y) = 1\). Here, \(h = h(-p)\) is the class number of the imaginary quadratic field \(\mathbb{Q}(\sqrt{-p})\). Using a combination of Galois representations associated with Frey-Hellegoaurch elliptic curves, tools from algebraic number theory, and explicit computations with the Magma computational package, they solve the equation under consideration. These results extend earlier findings of Chakraborty et al. concerning the equation of the form \(x^2 + p^3 = 4y^n\) [\textit{K. Chakraborty} et al., Publ. Math. Debr. 97, No. 3--4, 339--352 (2020; Zbl 1474.11090)].