We have previously considered continued fractions with “numerator” a positive integer N, which we refer to as cfN\documentclass[12pt]{minimal}
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\begin{document}$${\text {cf}}_N$$\end{document} expansions. In particular, let E be a positive integer that is not a perfect square. For N>1\documentclass[12pt]{minimal}
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\begin{document}$$N > 1$$\end{document}, E\documentclass[12pt]{minimal}
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\begin{document}$$\sqrt{E}$$\end{document} has infinitely many cfN\documentclass[12pt]{minimal}
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\begin{document}$${\text {cf}}_N$$\end{document} expansions. There is a natural notion of the “best” cfN\documentclass[12pt]{minimal}
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\begin{document}$${\text {cf}}_N$$\end{document} expansion of E\documentclass[12pt]{minimal}
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\begin{document}$$\sqrt{E}$$\end{document}. We have conjectured, based on extensive numerical evidence, that such a best expansion is not always periodic. From this evidence, it is difficult to predict for which N this expansion will be periodic. We show here that for any such E, there are infinitely many values of N for which this expansion is indeed periodic, more precisely, periodic of period 1 or 2, and we obtain formulas for a subset of these expansions in terms of solutions to Pell’s equation x2-Ey2=1\documentclass[12pt]{minimal}
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\begin{document}$$x^2 - Ey^2 = 1$$\end{document}.
