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arXiv · 1409.0446

Integer Complexity: Experimental and Analytical Results II

Juris Čerņenoks·Jānis Iraids·Mārtiņš Opmanis·Rihards Opmanis·Kārlis Podnieks
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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.

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Juris Čerņenoks, Jānis Iraids, Mārtiņš Opmanis, Rihards Opmanis, Kārlis Podnieks. 2014-09-01. Integer Complexity: Experimental and Analytical Results II. https://arxiv.org/abs/1409.0446

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