On the Diophantine equation \(by^2\pm 2=f(x)\) (Q2749263)

Let \(a,b\) be two positive integers, and let \(m>1\) be an odd integer. The reviewer [Chin. Sci. Bull. 35, 1227-1228 (1990; Zbl 0764.11020); ibid. 36, 275-278 (1991; Zbl 0741.11019)] gave all solutions to the Diophantine equations \(\frac{ax^m\pm 1}{abx\pm 1}=by^2\) and \(\frac{ax^m\pm 2}{abx\pm 2}=by^2\). In this paper, using Pell's equation method [see the reviewer's book: Introduction to Diophantine equations (Chinese), Harbin Press (1989; Zbl 0849.11029)], the author gives all solutions to Diophantine equations \(by^2\pm 2=f(x)\), where \(f(x)\in A, A=\{a(x(abx\pm 1))^m, a(x(abx\pm 2))^m,a(x(abx\pm 4))^m,ax(abx\pm 1)^m,ax(abx\pm 2)^m,ax(abx\pm 4)^m,a(abx\pm 1)x^m,a(abx\pm 2)x^m,a(abx\pm 4)x^m\}\).