Integer Complexity, Addition Chains, and Well-Ordering by Harry J. Altman A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in the University of Michigan 2014 Doctoral Committee: Professor Jeffrey C. Lagarias, Chair Professor Alexander Barvinok Professor Andreas R. Blass Associate Professor Kevin J. Compton Professor Martin J. Strauss For my grandparents ii Acknowledgments The author is grateful to Joshua Zelinsky, who originally suggested the subject of integer complexity, and together with whom much of the work in Chapter 2 was conducted. The author is grateful to Juan Arias de Reyna for much helpful discussion regard- ing integer complexity { improving notation, clarifying statements, and providing shorter proofs than the author's of the lower bound in Proposition 3.6.3 and the k = 1 case of Lemma 2.4.5. The author is grateful to his advisor Jeffrey C. Lagarias, for suggesting the ad- ditional topic of addition chains, for pointing out the relations to computational complexity issues, and for much helpful discussion. The author thanks in addition the following people: J. Iraids and K. Podnieks for providing much helpful numerical data; Andreas Blass, Paul Pollack, and Mike Bennett for suggesting references; E. H. Brooks for discussion regarding some of the proofs in Chapter 5; J. Heidi Soderstrom for help translating references into English; and A. Mishchenko for providing his LATEX files for his dissertation, from which the formatting of this dissertation has largely been copied. iii Preface A note to the reader: The bulk of this thesis, Chapters 2 through 5, were originally written as separate papers (the paper which became Chapter 2 was co-authored with J. Zelinsky). As such, each has its own individual abstract in addition to the over- all abstract, and the initial few sections of each chapter repeat much information from previous chapters. Appendix A was originally an appendix to Chapter 3, and Appendices B and C were originally appendices to Chapter 5. iv Contents Dedication ii Acknowledgments iii Preface iv List of Tables viii List of Figures ix List of Appendices x Abstract xi Chapter 1 Introduction 1 1.1 Notions of complexity for natural numbers . 1 1.2 Main results: Integer complexity . 8 1.3 Main results: Addition chains . 12 1.4 Other notions of complexity . 15 1.5 Plan of this thesis . 18 2 Numbers with Integer Complexity Close to the Lower Bound 19 2.1 Introduction . 19 2.2 Properties of the defect . 25 2.3 Good factorizations and solid numbers . 29 2.4 The Classification Method . 32 2.5 Determination of all elements of defect below a given bound r . 38 2.6 Applications . 49 3 Integer Complexity and Well-Ordering 54 3.1 Introduction . 54 3.2 Properties of the defect . 60 3.3 Stable defects and stable complexity . 63 3.4 Low-defect polynomials . 65 3.5 Facts from order theory and topology . 74 3.6 Well-ordering of defects . 78 v 3.7 Variants of the main theorem . 82 4 Addition Chains and Well-Ordering 87 4.1 Introduction . 87 4.2 Comparison of addition chain complexity and integer complexity . 95 4.3 The A-defect and A-stabilization . 97 4.4 Bit-counting in numbers of small defect . 101 4.5 Cutting and pasting well-ordered sets . 103 4.6 Well-ordering of defects . 105 4.7 Bounds on order type for small A-defect values . 109 4.8 Concluding Remarks . 112 5 Integer Complexity: Computational Methods and Results 113 5.1 Introduction . 113 5.2 The defect, stability, and low-defect polynomials . 123 5.3 Further notes on stabilization and stable complexity . 132 5.4 Low-defect expressions, the nesting ordering, and structure of low- defect polynomials . 133 5.5 The truncation operation . 149 5.6 Algorithms: Computing good coverings . 158 5.7 Algorithms: Computing stabilization length K(n) and stable com- plexity knkst .............................. 165 5.8 Results of computation . 174 6 Open problems and future research 177 6.1 Additional structure in the defect set . 177 6.2 Generalization to addition-multiplication chains . 180 6.3 Complexity based on a number other than 1 . 180 6.4 Further stabilization hypotheses . 181 6.5 Instability . 182 6.6 Counting problems . 184 6.7 Computability and complexity-theoretic problems . 185 6.8 Remaining computational problems . 186 Appendix A Conjectures of J. Arias de Reyna 187 B Good coverings of closed intervals 189 C Implementation notes 192 D Leaders with defect at most 1 195 vi References 196 vii List of Tables 5.1 Numbers that seem to have unusual drop patterns . 122 6.1 Numbers that seem to have unusual drop patterns . 183 D.1 Leaders of defect at most 1 . 195 viii List of Figures Figure 1.1 A tree for n = 11 using 8 ones and of height 3 . 18 Figure 3.1 Illustration of substitution into a low-defect polynomial . 57 Figure 3.2 Illustration of substitution of 30 into a low-defect polynomial . 69 Figure 5.1 Illustration of low-defect tree . 135 Figure 5.2 Two different trees yielding the polynomial 4x + 2 . 139 Figure 5.3 Illustration of bijection between variables and non-root vertices . 140 ix List of Appendices Appendix A: Conjectures of J. Arias de Reyna . 187 Appendix B: Good coverings of closed intervals . 189 Appendix C: Implementation notes . 192 Appendix D: Leaders with defect at most 1 . 195 x Abstract In this dissertation we consider two notions of the \complexity" of a natural num- ber, the first being addition chain length, and the second known simply as \integer complexity". The integer complexity of n, denoted knk, is the smallest number of 1's needed to write n using an arbitrary combination of addition and multiplication. It is known that knk ≥ 3 log3 n for all n. We consider the difference δ(n) := knk − 3 log3 n, which we call the defect of n. We consider the set of all defects { the set D := fδ(n): n 2 Ng: We show that, as a subset of the real numbers, D is well-ordered, with order type !!; we also show the same for several variants of this set. Moreover, we show that, for k ≥ 1 a natural number, D \ [0; k) has order type precisely !k. We also use the defect to prove stabilization results about knk. Specifically, for any n, there exists K = K(n) such that for k ≥ K, we have δ(3kn) = δ(3K n): We call K(n) the stabilization length of n. Finally, we provide a way of, given r > 0, computing all numbers n with δ(n) < r. We use this to show that the stabilization length K(n) is effectively computable. The algorithm is also, empirically, much faster than existing methods for computing k2kk, and we use it to prove that k2k3`k = 2k + 3` for 0 ≤ k ≤ 48 and ` ≥ 0, with k and ` not both 0. In parallel to our results for integer complexity, we also consider addition chain length. An addition chain for n is defined to be a sequence (a0; a1; : : : ; ar) such that a0 = 1, ar = n, and, for any k with 1 ≤ k ≤ r, there exist 0 ≤ i; j < k such that ak = ai + aj; the number r is called the length of the addition chain. The shortest xi length among addition chains for n, called the addition chain length of n, is denoted `(n). The number `(n) is always at least log2 n. ` We consider the difference δ (n) := `(n) − log2 n, which we call the addition-chain defect of n, and the set of all addition-chain defects ` ` D := fδ (n): n 2 Ng: We show that D ` is also a well-ordered set with order type !!. We also use the defect to prove stabilization results about `(n); specifically, for any n, there exists K0 = K0(n) such that for k ≥ K0, we have δ(2kn) = δ`(2K0 n): xii Chapter 1 Introduction 1.1 Notions of complexity for natural numbers In this dissertation we will consider the complexity of computing natural numbers under some simple computational models. When we speak of computing a natural number, we mean building it up in some finite number of steps from the number 1, which is the most basic of all natural numbers and generates all the others. There are various models of computation we could turn our attention to, but we will focus on two: One is known as \integer complexity" (Section 1.1.1), and the other is that of addition chains (Section 1.1.2). Some of the others will be briefly discussed in Section 1.4. In all these cases, we are discussing building up natural numbers from the number 1; what we vary is what tools are allowed. Of course, every natural number n can be written as the sum of n ones, and if we only allow the use of addition, it is impossible to do better, so this is not a very interesting model; something more is needed to allow shorter, less obvious ways of writing n. Thus we define the integer complexity of a natural number n to be the least number of 1's needed to write it using any combination of addition and multiplication, with the order of the operations specified using parentheses grouped in any legal nesting. For instance, 11 has complexity of 8, since it can be written using 8 ones as 11 = (1 + 1 + 1)(1 + 1 + 1) + 1 + 1; but not with any fewer. This notion was implicitly introduced in 1953 by Kurt Mahler and Jan Popken [38]; they actually considered an inverse function, the size of the largest number representable using k copies of the number 1.