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 socalled $P$algorithms  a family of "deterministic" algorithms for building representations of numbers.
 Publication:

arXiv eprints
 Pub Date:
 September 2014
 DOI:
 10.48550/arXiv.1409.0446
 arXiv:
 arXiv:1409.0446
 Bibcode:
 2014arXiv1409.0446C
 Keywords:

 Mathematics  Number Theory;
 11A63;
 11B99
 EPrint:
 21 pages, 3 figures