Le, Maohua2024-09-172024-09-172022-08-200019-3577https://doi.org/10.1016/j.indag.2022.04.005https://hdl.handle.net/11452/44818Let k be a positive integer. In this paper, using the modular approach, we prove that if k & EQUIV; 0 (mod 4), 30 < k < 724 and 2k -1 is an odd prime power, then under the GRH, the equation x2 + (2k -1)y = kz has only one positive integer solution (x, y, z) = (k - 1, 1, 2). The above results solve some difficult cases of Terai's conjecture concerning this equation.(c) 2022 Royal Dutch Mathematical Society (KWG).eninfo:eu-repo/semantics/closedAccessDiophantine equationsPointsPolynomial-exponential diophantine equationElliptic curveS-integral pointModular approachScience & technologyPhysical sciencesMathematicsArticle000849224700006992100033510.1016/j.indag.2022.04.005