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2019 Integer complexity: the integer defect
Harry Altman
Mosc. J. Comb. Number Theory 8(3): 193-217 (2019). DOI: 10.2140/moscow.2019.8.193


Define n to be the complexity of n, the smallest number of 1s needed to write n using an arbitrary combination of addition and multiplication. John Selfridge showed that n3log3n for all n, leading this author and Zelinsky to define the defect of n, δ(n), to be the difference n3log3n. Meanwhile, in the study of addition chains, it is common to consider s(n), the number of small steps of n, defined as (n)log2n, an integer quantity, where (n) is the length of the shortest addition chain for n. 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 well-ordering we considered in 2015, 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 (1989).


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Harry Altman. "Integer complexity: the integer defect." Mosc. J. Comb. Number Theory 8 (3) 193 - 217, 2019.


Received: 2 August 2018; Revised: 7 April 2019; Accepted: 29 April 2019; Published: 2019
First available in Project Euclid: 13 August 2019

zbMATH: 07095940
MathSciNet: MR3990804
Digital Object Identifier: 10.2140/moscow.2019.8.193

Primary: 11A67

Keywords: arithmetic formulas , integer complexity , well-ordering

Rights: Copyright © 2019 Mathematical Sciences Publishers


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Vol.8 • No. 3 • 2019
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