0000000000661559
AUTHOR
Juris ČErņenoks
Dažas neatrisinātas problēmas kombinatoriskajā ģeometrijā
Maģistra darbā risinātas dažas kombinatoriskās ģeometrijas problēmas par tetradiem un pakošanas problēma ar V-pentakubiem. Šo problēmu risināšanai ir izstrādātas speciālas datorprogrammas un iegūti vairāki jauni rezultāti: ir atrasti visi tetradi no polimino, kuru laukums nepārsniedz 60; iegūti paralēlskaldņu n x n x 1, n < 21, un 5 x 5 x 5 blīvākie pakojumi no V-pentakubiem. Atslēgvārdi: n-mino, pakošana, pentakubs, polimino, tetrads.
Integer Complexity: Experimental and Analytical Results
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 number of ones in the shortest expressions representing n. We arrive here very soon at the problems that are easy to formulate, but (it seems) extremely hard to solve. In this paper we represent our attempts to explore the field by means of experimental mathematics. Having computed the values of ||n|| up to 10^12 we present our observations. One of them (if true) implies that there is an infinite number of Sophie Germain primes, and even that there is an infinite number of Cunningham chains of len…
Integer Complexity: Experimental and Analytical Results II
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 $…
Integer Complexity: Experimental and Analytical Results II
We consider representing natural numbers by expressions using only 1’s, addition, multiplication and parentheses. Let \( \left\| n \right\| \) denote the minimum number of 1’s in the expressions representing \(n\). The logarithmic complexity \( \left\| n \right\| _{\log } \) is defined to be \({ \left\| n \right\| }/{\log _3 n}\). 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 ba…