# Integer Complexity: Experimental and Analytical Results II

@inproceedings{Cernenoks2015IntegerCE, title={Integer Complexity: Experimental and Analytical Results II}, author={Juris Cernenoks and Janis Iraids and Martins Opmanis and Rihards Opmanis and Karlis Podnieks}, booktitle={DCFS}, year={2015} }

We consider representing of natural numbers by expressions using 1's, addition, multiplication and parentheses. $\left\| n \right\|$ denotes the minimum number of 1's in the expressions representing $n$. The logarithmic complexity $\left\| n \right\|_{\log}$ is defined as $\left\| n \right\|/{\log_3 n}$. The values of $\left\| n \right\|_{\log}$ are located in the segment $[3, 4.755]$, but almost nothing is known with certainty about the structure of this "spectrum" (are the values dense…

## 10 Citations

### Integer Complexity: Algorithms and Computational Results

- Computer Science, MathematicsIntegers
- 2018

An algorithm for computing K(n) is described, and it is shown that the set of stable numbers is a computable set, and the defect of a number is defined by $\delta(n):=\|n\|-3\log_3 n$, building on the methods presented in [3].

### Algorithms for determining integer complexity

- Computer Science
- 2014

Three algorithms to compute the complexity of all natural numbers, each superior to the one in [11], are presented and it is shown that they run in time $O(N^\alpha)$ and space $N\log\log N)$.

### An Application of Markov Chain Analysis to Integer Complexity

- Computer Science, Mathematics
- 2015

Markov chain methods are used to analyze a large class of algorithms, including one found by David Bevan that improves the upper bound to f(n) \leq 3.52 \log_{3}{n}$$ on a set of logarithmic density one.

### On Algorithms to Calculate Integer Complexity

- Mathematics, Computer ScienceIntegers
- 2019

A method is discussed that provides a strong uniform bound on the number of summands that must be calculated for almost all $n$ and potential improvements to this algorithm are discussed.

### Digits of pi: limits to the seeming randomness

- Mathematics
- 2014

The decimal digits of $\pi$ are widely believed to behave like as statistically independent random variables taking the values $0, 1, 2, 3, 4, 5$, $6, 7, 8, 9$ with equal probabilities $1/10$.
In…

### Upper Bounds on Integer Complexity

- Mathematics
- 2022

This paper provides the first non-trivial upper bound for all n, where for all $n>1$ the authors have $||n|| \leq A \log n$ where $A = \frac{41}{\log 55296}$.

### Optimal Presentations of Mathematical Objects

- Mathematics
- 2018

We discuss the optimal presentations of mathematical objects under well defined symbol libraries. We shall examine what light our chosen symbol libraries and syntax shed upon the objects they…

### Some Conceptual and Measurement Aspects of Complexity, Chaos, and Randomness from Mathematical Point of View

- Computer Science
- 2018

This study aims to reveal and emphasize the role of mathematics in formulating the conceptual and measurement stages of complexity, chaos, and randomness in natural dynamical systems.

### Integer Complexity and Well-Ordering

- Mathematics
- 2013

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\| \ge…

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Three algorithms to compute the complexity of all natural numbers, each superior to the one in [11], are presented and it is shown that they run in time $O(N^\alpha)$ and space $N\log\log N)$.

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