No-three-in-line problem on a torus: Periodicity

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F19%3A10404004" target="_blank" >RIV/00216208:11320/19:10404004 - isvavai.cz</a>
Result on the web
<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=d__8apGqwG" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=d__8apGqwG</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1016/j.disc.2019.111611" target="_blank" >10.1016/j.disc.2019.111611</a>

Result language
angličtina
Original language name
No-three-in-line problem on a torus: Periodicity
Original language description
Let tau(m,n) denote the maximal number of points on the discrete torus (discrete toric grid) of sizes m x n with no three collinear points. The value tau(m,n) is known for the case where gcd(m, n) is prime. It is also known that tau(m,n) <= 2 gcd(m, n). In this paper we generalize some of the known tools for determining tau(m,n) and also show some new. Using these tools we prove that the sequence (tau(z,n))(n is an element of N) is periodic for all fixed z > 1. In general, we do not know the period; however, if z = p(a) for p prime, then we can bound it. We prove that tau(pa,p(a-1)p+2) = 2p(a) which implies that the period for the sequence is p(b), where b is at most (a - 1)p + 2.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2019
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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