Abstract. Define n to be the complexity of n, the smallest number of ones needed to write n using an arbitrary combination of addition and multiplication. John Selfridge showed that n ≥ 3 log 3 n for all n. Define the defect of n, denoted δ(n), to be n − 3 log 3 n. In this paper, we consider the set D := {δ(n) : n ≥ 1} of all defects. We show that as a subset of the real numbers, the set D is well-ordered, of order type ω ω . More specifically, for k ≥ 1 an integer, D ∩ [0, k) has order type ω k . We also consider some other sets related to D, and show that these too are well-ordered and have order type ω ω .

Define knk to be the complexity of n, the smallest number of ones needed to write n using an arbitrary combination of addition and multiplication. John Selfridge showed that knk 3 log 3 n for all n. Define the defect of n, denoted by ı.n/, to be knk 3 log 3 n; in this paper we present a method for classifying all n with ı.n/ < r for a given r. From this, we derive several consequences. We prove that k2 m 3 k k D 2m C 3k for m Ä 21 with m and k not both zero, and present a method that can, with more computation, potentially prove the same for larger m. Furthermore, defining A r .x/ to be the number of n with ı.n/ < r and n Ä x, we prove that A r .x/ D ‚ r ..log x/ brcC1 /, allowing us to conclude that the values of knk 3 log 3 n can be arbitrarily large.

Abstract. Define n to be the complexity of n, the smallest number of ones needed to write n using an arbitrary combination of addition and multiplication. John Selfridge showed that n ≥ 3 log 3 n for all n. Based on this, this author and Zelinsky defined [4] the "defect" of n, δ(n) := n − 3 log 3 n, and this author showed that the set of all defects is a well-ordered subset of the real numbers [1]. This was accomplished by showing that for a fixed real number r, there is a finite set S of polynomials called "low-defect polynomials" such that for any n with δ(n) < r, n has the form f (3 k 1 , . . . , 3 kr )3 k r+1 for some f ∈ S. However, using the polynomials produced by this method, many extraneous n with δ(n) ≥ r would also be represented. In this paper we show how to remedy this and modify S so as to represent precisely the n with δ(n) < r and remove anything extraneous. Since the same polynomial can represent both n with δ(n) < r and n with δ(n) ≥ r, this is not a matter of simply excising the appropriate polynomials, but requires "truncating" the polynomials to form new ones.

Abstract. An addition chain for n is defined to be a sequence (a 0 , a 1 , . . . , ar) such that a 0 = 1, ar = n, and, for any 1 ≤ k ≤ r, there exist 0 ≤ i, j < k such that a k = a i + a j ; the number r is called the length of the addition chain. The shortest length among addition chains for n, called the addition chain length of n, is denoted ℓ(n). The number ℓ(n) is always at least log 2 n; in this paper we consider the difference δ ℓ (n) := ℓ(n) − log 2 n, which we call the addition chain defect. First we use this notion to show that for any n, there exists K such that for any k ≥ K, we have ℓ(2 k n) = ℓ(2 K n) + (k − K). The main result is that the set of values of δ ℓ is a well-ordered subset of [0, ∞), with order type ω ω . The results obtained here are analogous to the results for integer complexity obtained in [1] and [3]. We also prove similar well-ordering results for restricted forms of addition chain length, such as star chain length and Hansen chain length.

There are two well‐known ways of doing arithmetic with ordinal numbers: the “ordinary” addition, multiplication, and exponentiation, which are defined by transfinite iteration; and the “natural” (or “Hessenberg”) addition and multiplication (denoted ⊕ and ⊗), each satisfying its own set of algebraic laws. In 1909, Jacobsthal considered a third, intermediate way of multiplying ordinals (denoted × ), defined by transfinite iteration of natural addition, as well as the notion of exponentiation defined by transfinite iteration of his multiplication, which we denote α×β. (Jacobsthal's multiplication was later rediscovered by Conway.) Jacobsthal showed these operations too obeyed algebraic laws. In this paper, we pick up where Jacobsthal left off by considering the notion of exponentiation obtained by transfinitely iterating natural multiplication instead; we shall denote this α⊗β. We show that α⊗(β⊕γ)=false(α⊗βfalse)⊗false(α⊗γfalse) and that α⊗(β×γ)=(α⊗β)⊗γ; note the use of Jacobsthal's multiplication in the latter. We also demonstrate the impossibility of defining a “natural exponentiation” satisfying reasonable algebraic laws.