A NOTE ON UNITS OF REAL QUADRATIC FIELDS

For a positive square-free integer d, let t(d) and u(d) be positive integers such that is an element of(d) = t(d)+u(d)root d/sigma is the fundamental unit of the real quadratic field Q(root d), where sigma = 2 if d equivalent to 1 (mod 4) and sigma = 1 otherwise. For a given positive integer l and a palindromic sequence of positive integers a(1), . . . ,a(l-1) we define the set S(l; a(1), . . . ,a(l-1)) := {d is an element of Z vertical bar d > 0, root d = [a(0), <(a(1), . . . , a(l-1) , 2a(0))]}. We prove that u(d) < d for all square-free integer d is an element of S(l; a(1), . . . ,a(l-1)) with one possible exception and apply it to Ankeny-Artin-Chowla conjecture and Mordell conjecture.