On units of real quadratic fields

AbstractLet ∈ = (t + u(d)2/2 be a unit of Q((d)1/2), whose norm = 1. We investigate the properties of the factors of the number un which is defined by ∈n = (tn + un(d)1/2)/2.Those properties are applied to the theory of the fundamental unit of Q((d)1/2). The main result is as follows. Take some unit η = (a + b(d)12)/2 (>1) whose norm = 1. If the number b satisfies a certain condition, then η = ∈02n for some n, where ∈0 is the fundamental unit of Q((d)1/2)