• DOI:10.1007/978-3-319-19225-3_5
• Corpus ID: 38583739

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}
}

• Juris Cernenoks, Janis Iraids, +2 authors K. Podnieks
• Published in
  Workshop on Descriptional…
  1 September 2014
• Mathematics

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 somewhere in the segment etc.). We establish a connection between this problem and another difficult problem: the seemingly "almost random" behaviour of digits in the base 3 representations of the numbers $2^n$.
We consider also representing of natural numbers by expressions that include subtraction, and the so-called $P$-algorithms - a family of "deterministic" algorithms for building representations of numbers.