Ambiguous solutions of a Pell equation (Q6102096)

Let \(D\) be a square-free integer with \(D>1\) and \(N\) a nonzero integer. Consider the generalized Pell equation \[ X^2-DY^2=N.\tag{1} \] It is known that the set of all integer solutions of (1) is divided into only finitely many classes of solutions, each of which has the least positive solution \((X,Y)=(x,y)\), called \textit{a fundamental solution}. In fact, when we denote by \((u_1,v_1)\) the fundamental solution of the Pell equation \(U^2-DV^2=1\), the explicit ranges for \(x\) and \(y\) are given by the following: \begin{itemize} \item If \(N>0\), then \[ 0<|x|\le\sqrt{\frac{(u_1+1)N}{2}},\quad 0 \le y\le \sqrt{\frac{(u_1-1)N}{2D}}.\tag{>} \] \item If \(N<0\), then \[ 0\le |x|\le\sqrt{\frac{(u_1-1)|N|}{2}},\quad 0<y\le \sqrt{\frac{(u_1+1)|N|}{2D}}.\tag{<} \] \end{itemize} We call a class of solutions \((x,y)\) \textit{ambiguous} if \((x,y)\) and \((-x,y)\) belong to the same class of solutions. The main theorem in this paper under review asserts the following: Theorem 6. Let \((x,y)\) with \(xy\ne 0\) be a fundamental solution of equation (1). \begin{itemize} \item[(i)] \((x,y)\) defines an ambiguous class if and only if \(x\) and \(y\) attain the upper bounds (>) or (<) depending on \(N>0\) or \(N<0\), respectively. \item[(ii)] In case \((x,y)\) defines an ambiguous class, all solutions \((x_n,y_n)\) with \(n\) integers in the class are given by \[ x_n+y_n\sqrt{D}=\frac{\pm(x+y\sqrt{D}\,)^{2n+1}}{|N|^n}. \] \end{itemize} As noted in Acknowledgements, assertion (i) in the theorem above was proved in Theorem 2 of the unpublished paper [\textit{K. Matthews} and \textit{J. Robertson}, ``On the converse of a theorem of Nagell and Tchebichef'', Preprint, \url{http://www.numbertheory.org/PDFS/nagell2.pdf}]. As a corollary to the theorem, the authors remark that a similar assertion holds for the Diophantine equation of the form \[ aX^2-bY^2=\pm c,\tag{abc} \] where \(a,b,c\) are positive integers with \(a>1\), \(b>1\) and \(ab\) square-free, by noting that the integer solutions of (abc) are in one-to-one correspondence with the integer solutions of \(X^2-abY^2=\pm ac\). This remark proves a result in [\textit{D. T. Walker}, Am. Math. Mon. 74, 504--513 (1967; Zbl 0154.29604)], which can be regarded as the case where \(c=1\) in equation (abc).