On polynomials solutions of quadratic Diophantine equations (Q2893948)

The author considers the integer solutions to the Diophantine equation NEWLINE\[NEWLINE E: X^{2}-(P^{2}-P)Y^{2}-(4P-2)X+(4P^{2}-4P)Y=0 NEWLINE\]NEWLINE where \(P=P(t)\) is a polynomial in \(\mathbb{Z}[X]-\{0,1\}\).NEWLINENEWLINEHe transforms \(E\) into the Pell equation NEWLINE\[NEWLINE \widetilde{E}: U^{2}-(P(t)^{2}-P(t))V^{2}=1 NEWLINE\]NEWLINE and in Theorem 2.1 he derives some results related to integer solutions to \(\widetilde{E}\). Later he retransfers all results from \(\widetilde{E}\) to \(E\) in Theorem 2.2.NEWLINENEWLINEIn fact these two theorems are proved in the same manner as in the previous work written by \textit{A. Özkoç} and the reviewer [Quadratic Diophantine equation \(x^{2}-(t^{2}-t)y^{2}-(4t-2)x+(4t^{2}-4t)y=0\), Bull. Malays. Math. Sci. Soc. (2) 33, No. 2, 273--280 (2010; Zbl 1198.11032)] just by taking \(t\rightarrow P\).