| Title:
|
Construction of Šindel sequences (English) |
| Author:
|
Křížek, Michal |
| Author:
|
Šolcová, Alena |
| Author:
|
Somer, Lawrence |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
48 |
| Issue:
|
3 |
| Year:
|
2007 |
| Pages:
|
373-388 |
| . |
| Category:
|
math |
| . |
| Summary:
|
We found that there is a remarkable relationship between the triangular numbers $T_k$ and the astronomical clock (horologe) of Prague. We introduce Šindel sequences $\{a_i\}\subset \Bbb N$ of natural numbers as those periodic sequences with period $p$ that satisfy the following condition: for any $k\in\Bbb N$ there exists $n\in\Bbb N$ such that $T_k=a_1+\cdots+a_n$. We shall see that this condition guarantees a functioning of the bellworks, which is controlled by the horologe. We give a necessary and sufficient condition for a periodic sequence to be a Šindel sequence. We also present an algorithm which produces the so-called primitive Šindel sequence, which is uniquely determined for a given $s=a_1+\cdots+a_p$. (English) |
| Keyword:
|
Jacobi symbol |
| Keyword:
|
quadratic nonresidue |
| Keyword:
|
clock sequence |
| Keyword:
|
primitive Šindel sequences |
| Keyword:
|
Chinese remainder theorem |
| Keyword:
|
Dirichlet's theorem |
| MSC:
|
01A40 |
| MSC:
|
11A07 |
| MSC:
|
11A51 |
| MSC:
|
11B83 |
| idZBL:
|
Zbl 1174.11029 |
| idMR:
|
MR2374121 |
| . |
| Date available:
|
2009-05-05T17:03:33Z |
| Last updated:
|
2012-05-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119666 |
| . |
| Reference:
|
[1] Burton D.M.: Elementary Number Theory.fourth edition, McGraw-Hill, New York (1989, 1998). MR 0990017 |
| Reference:
|
[2] Horský Z.: The Astronomical Clock of Prague (in Czech).Panorama, Prague (1988). |
| Reference:
|
[3] Křížek M., Luca F., Somer L.: 17 Lectures on Fermat Numbers: From Number Theory to Geometry.CMS Books in Mathematics, vol. 9, Springer New York (2001). Zbl 1010.11002, MR 1866957 |
| Reference:
|
[4] Niven I., Zuckerman H.S., Montgomery H.L.: An Introduction to the Theory of Numbers.fifth edition, John Wiley & Sons, New York (1991). Zbl 0742.11001, MR 1083765 |
| Reference:
|
[5] Sloane N.J.A.: My favorite integer sequences.arXiv: math. C0/0207175v1 (2002), 1-28. Zbl 1049.11026, MR 1843083 |
| Reference:
|
[6] Tattersall J.J.: Elementary Number Theory in Nine Chapters.second edition, Cambridge Univ. Press, Cambridge (2005). Zbl 1071.11002, MR 2156483 |
| Reference:
|
[7] HASH(0x9391e98): http://www.research.att.com/\char` njas/sequences/.. |
| . |
