Article
Full entry |
PDF
(0.5 MB)
Feedback
PDF
(0.5 MB)
Feedback
Keywords:
Jacobi symbol; quadratic nonresidue; clock sequence; primitive Šindel sequences; Chinese remainder theorem; Dirichlet's theorem
Jacobi symbol; quadratic nonresidue; clock sequence; primitive Šindel sequences; Chinese remainder theorem; Dirichlet's theorem
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 $s=a_1+\cdots+a_p$.
References:
[1] Burton D.M.: Elementary Number Theory. fourth edition, McGraw-Hill, New York (1989, 1998). MR 0990017
[2] Horský Z.: The Astronomical Clock of Prague (in Czech). Panorama, Prague (1988).
[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). MR 1866957 | Zbl 1010.11002
[4] Niven I., Zuckerman H.S., Montgomery H.L.: An Introduction to the Theory of Numbers. fifth edition, John Wiley & Sons, New York (1991). MR 1083765 | Zbl 0742.11001
[5] Sloane N.J.A.: My favorite integer sequences. arXiv: math. C0/0207175v1 (2002), 1-28. MR 1843083 | Zbl 1049.11026
[6] Tattersall J.J.: Elementary Number Theory in Nine Chapters. second edition, Cambridge Univ. Press, Cambridge (2005). MR 2156483 | Zbl 1071.11002
[7] HASH(0x9391e98): http://www.research.att.com/\char` njas/sequences/.
Search
Partner of
