Continued Radicals

and consider continued radicals of form limn→∞ Sn = √ a1, a2, a3, . . . . Convergence criteria for continued radicals are given in [2], and [3]. We consider the sets S(M) of real numbers which are representable as a continued radical whose terms a1, a2, . . . are all from a finite set M = {m1,m2, . . . ,mp} ⊆ N where 0 < m1 < m2 < · · · < mp. For any nonnegative number n, √ n, n, n, . . . converges to φn ≡ 1+ √ 4n+1 2 . It is easy to see that φn = k ∈ N if and only if n = k(k− 1) for some integer k ≥ 2. If S(M) is to be an interval, then to insure that no gaps occur in S(M), it is necessary that the largest value representable with a1 = mi equal or exceed the smallest value representable with a1 = mi+1 (for i = 1, . . . , p−1). That is, it is necessary that √ mi,mp,mp,mp, . . . ≥ √ mi+1,m1,m1,m1, . . . ∀i ∈ {1, . . . , p− 1},