Diophantine equations with at most one positive solution (Q1360927)

Let \(D\) be a square-free positive integer. In 1942, \textit{W. Ljunggren} [Arch. Math. Naturvid. B 45, 1-12 (1942; Zbl 0026.29604)] showed that the equation \[ (*) x^4-Dy^2= 1,\;x,y \in\mathbb{N}, \] has at most two solutions \((x,y)\). In 1989, \textit{M. H. Le} [Proc. Am. Math. Soc. 107, No. 1, 27-34 (1989; Zbl 0681.10012)] proved that if \(D>C\), where \(C\) is an effectively computable absolute constant, then (*) has at most one solution \((x,y)\). The proofs depend on the estimates for linear forms in two logarithms. Moreover, the reviewer conjectured that if \(D\neq 1785\), then (*) has at most one solution \((x,y)\). In this paper, using the same method, the author verifies the above-mentioned conjecture. \{Reviewer's remark: A similar result has been proved by \textit{J. H. E. Cohn} [Acta Arith. 78, No. 4, 401-403 (1997; Zbl 0870.11018)]\}.