Upper bounds on integer complexity
J Zelinsky - arxiv preprint arxiv:2211.02995, 2022 - arxiv.org
Define $|| n|| $ to be the\emph {complexity} of $ n $, which is the smallest number of $1 $ s
needed to write $ n $ using an arbitrary combination of addition and multiplication. John …
needed to write $ n $ using an arbitrary combination of addition and multiplication. John …
Numbers with integer complexity close to the lower bound
H Altman, J Zelinsky - Integers, 2012 - degruyter.com
Define 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 for all n …
arbitrary combination of addition and multiplication. John Selfridge showed that for all n …
Integer complexity and well-ordering
H Altman - Michigan Mathematical Journal, 2015 - projecteuclid.org
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 log3 n …
arbitrary combination of addition and multiplication. John Selfridge showed that n≥ 3 log3 n …
Integer complexity: Representing numbers of bounded defect
H Altman - Theoretical Computer Science, 2016 - Elsevier
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‖≥ …
an arbitrary combination of addition and multiplication. John Selfridge showed that‖ n‖≥ …
Integer complexity: Algorithms and computational results
H Altman - arxiv preprint arxiv:1606.03635, 2016 - arxiv.org
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. Define $ n $ to be stable if for …
$ using an arbitrary combination of addition and multiplication. Define $ n $ to be stable if for …
On algorithms to calculate integer complexity
We consider a problem first proposed by Mahler and Popken in 1953 and later developed
by Coppersmith, Erd\H {o} s, Guy, Isbell, Selfridge, and others. Let $ f (n) $ be the complexity …
by Coppersmith, Erd\H {o} s, Guy, Isbell, Selfridge, and others. Let $ f (n) $ be the complexity …
Algorithms for determining integer complexity
JA de Reyna, J Van de Lune - arxiv preprint arxiv:1404.2183, 2014 - arxiv.org
We present three algorithms to compute the complexity $\Vert n\Vert $ of all natural numbers
$ n\le N $. The first of them is a brute force algorithm, computing all these complexities in …
$ n\le N $. The first of them is a brute force algorithm, computing all these complexities in …
A short note on integer complexity
S Steinerberger - Contributions to Discrete mathematics, 2014 - cdm.ucalgary.ca
A SHORT NOTE ON INTEGER COMPLEXITY 1. Introduction The following problem of
Mahler and Popken [6] dates back to 1953: What is th Page 1 Volume 9, Number 1, Pages …
Mahler and Popken [6] dates back to 1953: What is th Page 1 Volume 9, Number 1, Pages …
Integer complexity, addition chains, and well-ordering
HJ Altman - 2014 - search.proquest.com
Integer Complexity, Addition Chains, and Well-Ordering Page 1 Integer Complexity, Addition
Chains, and Well-Ordering by Harry J. Altman A dissertation submitted in partial fulfillment of the …
Chains, and Well-Ordering by Harry J. Altman A dissertation submitted in partial fulfillment of the …
Integer complexity: Experimental and analytical results II
We consider representing natural numbers by expressions using only 1's, addition,
multiplication and parentheses. Let\left ‖ n\right ‖ denote the minimum number of 1's in the …
multiplication and parentheses. Let\left ‖ n\right ‖ denote the minimum number of 1's in the …