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

In this paper, we continue the study of Mutlu and Soydan (Int. J. Number Theory 20(5) (2024) 1195-1218), regarding the Diophantine equation x(2) + p(s) = 2 (R) y(n) for certain primes p. Theorem 1.1 of that paper missed the elimination of some non-rational newforms, and this is corrected in the corresponding Corollary 1.2 of this paper. Further, as new results, we solve some seventh degree Thue-Mahler equations allowing us to extend and improve our results by solving the case n = 7. Our main tools include the known results from the modularity of Galois representations associated with Frey-Hellegouarch elliptic curves (the strategy of Bennett-Skinner) and a Thue-Mahler solver which was improved by Gherga and Siksek.