EXISTENCE OF THE CONTINUED FRACTIONS OF d AND ITS APPLICATIONS

It is well known that the continued fraction expansion of root d has the form [a(0), a(1), . . . , a(l-1), 2a(0)] and a(1), . . . , a(l-1) is a palindromic sequence of positive integers. For a given positive integer l and a palindromic sequence of positive integers a(1), . . . , all(-1), we define the set S(l; a(1), ..., a(l-1)) := {d is an element of Z vertical bar d > 0, v root d = [a(0), a(1),..., a(l-1), 2a(0)], where a(0) = [root d]}. In this paper, we completely determine when S(l; a(1), ... , a(l-1)) is not empty in the case that l is 4, 5, 6, or 7. We also give similar results for (1 + d)/2. For the case that l is 4, 5, or 6, we explicitly describe the v/ fundamental units of the real quadratic field Q(root d). Finally, we apply our results to the Mordell conjecture for the fundamental units of Q(root d).