On the Diophantine equation \(X^2-(p^{2m}+1)Y^6=-p^{2m}\) (Q1959785)

Let \(p\) be a prime and \(m\) a positive integer. The authors prove the following Theorem: The equation in the title has at most four solutions in positive integers \((X, Y)\). In Section 2, like \textit{P. Yuan} [Sci. China, Ser. A 40, No. 10, 1045--1051 (1997; Zbl 0908.11027), J. Number Theory 102, No. 1, 1--10 (2003; Zbl 1095.11037)] and \textit{P. M. Voutier} [Acta Arith. 143, No. 2, 101-144 (2010; Zbl 1292.11083)], they give some effective irrationality measures for numbers over imaginary quadratic fields. The authors improve Yuan's result [loc. cit.] and adapt it to degree 6. In Section 3 they prove some preliminary results related to the solutions and the final section contains the proof of the authors' theorem.