Construction of Šindel sequences

Křížek, Michal, Šolcová, Alena, and Somer, Lawrence. "Construction of Šindel sequences." Commentationes Mathematicae Universitatis Carolinae 48.3 (2007): 373-388. <http://eudml.org/doc/250191>.

@article{Křížek2007,
abstract = {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 $\lbrace a_i\rbrace \subset \mathbb \{N\}$ of natural numbers as those periodic sequences with period $p$ that satisfy the following condition: for any $k\in \mathbb \{N\}$ there exists $n\in \mathbb \{N\}$ 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 $s=a_1+\cdots +a_p$.},
author = {Křížek, Michal, Šolcová, Alena, Somer, Lawrence},
journal = {Commentationes Mathematicae Universitatis Carolinae},
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},
language = {eng},
number = {3},
pages = {373-388},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Construction of Šindel sequences},
url = {http://eudml.org/doc/250191},
volume = {48},
year = {2007},
}

TY - JOUR
AU - Křížek, Michal
AU - Šolcová, Alena
AU - Somer, Lawrence
TI - Construction of Šindel sequences
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2007
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 48
IS - 3
SP - 373
EP - 388
AB - 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 $\lbrace a_i\rbrace \subset \mathbb {N}$ of natural numbers as those periodic sequences with period $p$ that satisfy the following condition: for any $k\in \mathbb {N}$ there exists $n\in \mathbb {N}$ 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 $s=a_1+\cdots +a_p$.
LA - eng
KW - 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
UR - http://eudml.org/doc/250191
ER -