Integer complexity: the integer defect

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, leading this author and Zelinsky to define the defect of n, δ(n), to be the difference n − 3 log 3 n. Meanwhile, in the study of addition chains, it is common to consider s(n), the number of small steps of n, defined as ℓ(n) − ⌊log 2 n⌋, an integer quantity. So here we analogously define D(n), the integer defect of n, an integer version of δ(n) analogous to s(n). Note that D(n) is not the same as ⌈δ(n)⌉.We show that D(n) has additional meaning in terms of the defect wellordering considered in [3], in that D(n) indicates which powers of ω the quantity δ(n) lies between when one restricts to n with n lying in a specified congruence class modulo 3. We also determine all numbers n with D(n) ≤ 1, and use this to generalize a result of Rawsthorne [18].

Integer complexity, stability, and self-similarity

“…The proofs of Proposition 4.3 and Theorem 1.9 uses a similar idea to the earlier proofs of (1.6), proven in [3], and (1.8), proven in [5]. Those were both also cases where, to get a lower bound on the order type of a particular set of defects, it was shown that particular low-defect polynomials "usually" yield numbers which have the complexity one would expect; and this in turn was proven by noting that unless this were so, this would violate a known upper bound on the order type of a smaller set of defects.…”

“…Why do we call such polynomials "substantial"? In making this definition, we build upon the technique used earlier in [5]. Suppose that η ∈ D, and suppose it takes the form δ(f ) for some low-defect polynomial f (which it always will, but which we of course have not yet proven at this point in the paper).…”

“…Note that there is no equivalent of the modulo-3 results for addition chains, due to our basic inequality being ℓ(2n) ≤ ℓ(n) + 1, rather than 3n ≤ n + 3; see Section 1E of [5] for a more detailed discussion of this point. 1.7.…”

“…( 6) n st ≤ n , with equality if and only if n is stable. (7) 8) are Proposition 2.7 from [5]. Part ( 9) is Proposition 9 from Section 7 of [1].…”

Integrative Mathematics Learning: Study of Hadiths on Number

“…Numbers are one of the basic knowledge in mathematics. Numbers are a mathematical concept used for solving and measuring (Altman, 2019;Davarpanah & Mirshekari, 2019;Dudley & Dudley, 2012;Tristianto, Linawati, & Susanto, 2018). Symbols or symbols used to represent a number are called numbers or symbols.…”