Note on a theorem of Rockett and Szüsz on a Diophantine equation \(x^2-dy^2=N\) (Q2717610)

The aim of the paper is to correct an inaccuracy which occurred in the joint paper of \textit{A. Rockett} and the reviewer [Acta Arith. 47, 347-350 (1986; Zbl 0561.10013)]. Our result was that the solutions of the Pellian equation \((*)\) \(x^2- dy^2= N\) can be characterized by the ``Ostrowski algorithm'' \((**)\) \(x= c_{k+1}A_k+ c_{k+2}A_{k+1}+\dots\), \(y= c_{k+1}B_k+ c_{k+2}B_{k+1}+\dots\), where the \(A_j\), resp. \(B_j\) are the numerators, resp. denominators of the convergents of the regular continued fraction of \(\sqrt{d}\) (\(d\) is supposed to be a natural number, not a square). Our (inaccurate) statement was that the only solutions of \((*)\) have the form NEWLINE\[NEWLINEN= \sum_{j=k}^l c_{j+1}(A_j-B_j\sqrt{d}) \sum_{j=k}^l c_{j+1}(A_j+B_j\sqrt{d}), \tag \(***\) NEWLINE\]NEWLINE where for each \(j\) we have \(0\leq c_{j+1}\leq a_{j+1}\) and \(c_{j+1}= a_{j+1}\) implies \(c_j=0\); here the \(a_{j+1}\) are the partial quotients of the regular continued fraction expansions of \(\sqrt{d}\) and the ``coefficients'' \(c_{j+1}\) depend only on \(N\). NEWLINENEWLINENEWLINEThe author found counterexamples to \((***)\) but showed that it is true if \(x\) in \((**)\) is sufficiently large. The method of the proof is identical with that of our paper; it is based on the Ostrowski algorithm just described.