Integer Complexity: Experimental and Analytical Results II
@inproceedings{Cernenoks2014IntegerCE, title={Integer Complexity: Experimental and Analytical Results II}, author={Juris Cernenoks and Janis Iraids and Martins Opmanis and Rihards Opmanis and Karlis Podnieks}, booktitle={Workshop on Descriptional Complexity of Formal Systems}, year={2014} }
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
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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].
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- 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
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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
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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…
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- 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.
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 $A = \frac{41}{\log 55296}$.
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…
References
SHOWING 1-10 OF 20 REFERENCES
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 $N\log\log N)$.
Integer Complexity: Experimental and Analytical Results
- Mathematics
- 2012
We consider representing of natural numbers by arithmetical expressions using ones, addition, multiplication and parentheses. The (integer) complexity of n -- denoted by ||n|| -- is defined as the…
Integer Complexity, Addition Chains, and Well-Ordering.
- Mathematics, Computer Science
- 2014
Two notions of the “complexity” of a natural number are considered, the first being addition chain length, and the second known simply as “integer complexity”, which is the smallest number of 1’s needed to write n using an arbitrary combination of addition and multiplication.
"How many $1$'s are needed?" revisited
- Computer Science
- 2014
We present a rigorous and relatively fast method for the computation of the "complexity" of a natural number (sequence A005245), and answer some "old and new" questions related to the question in the…
On the representation of an integer in two different bases.
- Mathematics
- 1980
In 1970 Senge and Strauss [4] proved that the number of integers, the sum of whose digits in each of the bases a and b lies below a fixed bound, is fmite if and only log if is irrational. Their…
Numbers with Integer Complexity Close to the Lower Bound
- MathematicsIntegers
- 2012
This paper presents a method for classifying all n with for a given r, and proves that for with m and k not both zero, and present a method that can, with more computation, potentially prove the same for larger m.
The On-Line Encyclopedia of Integer Sequences
- Computer ScienceElectron. J. Comb.
- 1994
The On-Line Encyclopedia of Integer Sequences (or OEIS) is a database of some 130000 number sequences which serves as a dictionary, to tell the user what is known about a particular sequence and is widely used.
A short note on integer complexity
- MathematicsContributions Discret. Math.
- 2014
This short note gives a new, constructive upper bound on the smallest number of 1's needed in conjunction with arbitrarily many +, *, and parentheses to write an integer n for generic integers n.
Champernowne’s Number, Strong Normality, and the X Chromosome
- Mathematics
- 2013
Champernowne’s number is the best-known example of a normal number, but its digits are far from random. The sequence of nucleotides in the human X chromosome appears nonrandom in a similar way. We…
Walking on Real Numbers
- Business
- 2013
Supported in part by the Director, Office of Computational and Technology Research, Division of Mathematical, Information, and Computational Sciences of the U.S. Department of Energy, under contract…