Periodicity of certain generalized continued fractions

We have previously considered continued fractions with “numerator” a positive integer N , which we refer to as cf N expansions. In particular, let E be a positive integer that is not a perfect square. For N > 1 , E has infinitely many cf N expansions. There is a natural notion of the “best” cf N expansion of E . 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 x 2 - E y 2 = 1 .