Integer Complexity: Experimental and Analytical Results II
Abstract
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- September 2014
- DOI:
- 10.48550/arXiv.1409.0446
- arXiv:
- arXiv:1409.0446
- Bibcode:
- 2014arXiv1409.0446C
- Keywords:
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- Mathematics - Number Theory;
- 11A63;
- 11B99
- E-Print:
- 21 pages, 3 figures