# On a connection of number theory with graph theory

Czechoslovak Mathematical Journal (2004)

- Volume: 54, Issue: 2, page 465-485
- ISSN: 0011-4642

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topSomer, Lawrence, and Křížek, Michal. "On a connection of number theory with graph theory." Czechoslovak Mathematical Journal 54.2 (2004): 465-485. <http://eudml.org/doc/30876>.

@article{Somer2004,

abstract = {We assign to each positive integer $n$ a digraph whose set of vertices is $H=\lbrace 0,1,\dots ,n-1\rbrace $ and for which there is a directed edge from $a\in H$ to $b\in H$ if $a^2\equiv b\hspace\{4.44443pt\}(mod \; n)$. We establish necessary and sufficient conditions for the existence of isolated fixed points. We also examine when the digraph is semiregular. Moreover, we present simple conditions for the number of components and length of cycles. Two new necessary and sufficient conditions for the compositeness of Fermat numbers are also introduced.},

author = {Somer, Lawrence, Křížek, Michal},

journal = {Czechoslovak Mathematical Journal},

keywords = {Fermat numbers; Chinese remainder theorem; primality; group theory; digraphs; Fermat numbers; Chinese remainder theorem; primality; group theory; digraphs},

language = {eng},

number = {2},

pages = {465-485},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {On a connection of number theory with graph theory},

url = {http://eudml.org/doc/30876},

volume = {54},

year = {2004},

}

TY - JOUR

AU - Somer, Lawrence

AU - Křížek, Michal

TI - On a connection of number theory with graph theory

JO - Czechoslovak Mathematical Journal

PY - 2004

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 54

IS - 2

SP - 465

EP - 485

AB - We assign to each positive integer $n$ a digraph whose set of vertices is $H=\lbrace 0,1,\dots ,n-1\rbrace $ and for which there is a directed edge from $a\in H$ to $b\in H$ if $a^2\equiv b\hspace{4.44443pt}(mod \; n)$. We establish necessary and sufficient conditions for the existence of isolated fixed points. We also examine when the digraph is semiregular. Moreover, we present simple conditions for the number of components and length of cycles. Two new necessary and sufficient conditions for the compositeness of Fermat numbers are also introduced.

LA - eng

KW - Fermat numbers; Chinese remainder theorem; primality; group theory; digraphs; Fermat numbers; Chinese remainder theorem; primality; group theory; digraphs

UR - http://eudml.org/doc/30876

ER -

## References

top- 10.2307/2315605, Amer. Math. Monthly 74 (1967), 152–156. (1967) Zbl0163.02605MR0207824DOI10.2307/2315605
- 10.1090/S0002-9904-1910-01892-9, Bull. Amer. Math. Soc. 16 (1910), 232–238. (1910) MR1558896DOI10.1090/S0002-9904-1910-01892-9
- Graphs and Digraphs (Third edition), Chapman & Hall, London, 1996. (1996) MR1408678
- Applications d’un corps fini dans lui-même, Dissertation, Univ. de Rennes I, 1984. (1984) MR0782298
- 10.1016/0012-365X(86)90024-5, Discrete Math. 61 (1986), 21–26. (1986) MR0850926DOI10.1016/0012-365X(86)90024-5
- Graph Theory, Addison-Wesley Publ. Company, London, 1969. (1969) Zbl0196.27202MR0256911
- Egy binom kongruenciáról, Az Egi Ho Si Mihn Tanárképzó Fóiskola füzetei (1978), 457–464. (1978)
- 17 Lectures on the Fermat Numbers. From Number Theory to Geometry, Springer-Verlag, New York, 2001. (2001) MR1866957
- A necessary and sufficient condition for the primality of Fermat numbers, Math. Bohem. 126 (2001), 541–549. (2001) MR1970256
- Discrete Iterations. Springer Series in Comput. Math. Vol. 6, Springer-Verlag, Berlin, 1986. (1986) MR0851186
- 10.1016/0012-365X(94)00250-M, Discrete Math. 148 (1996), 317–324. (1996) Zbl0843.05048MR1368298DOI10.1016/0012-365X(94)00250-M
- A discrete iteration in number theory, BDTF Tud. Közl. 8 (1992), 71–91. (Hungarian) (1992) Zbl0801.11011

## Citations in EuDML Documents

top- J. Skowronek-Kaziów, Properties of digraphs connected with some congruence relations
- Lawrence Somer, Michal Křížek, On semiregular digraphs of the congruence ${x}^{k}\equiv y\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}n)$
- Walter Carlip, Martina Mincheva, Symmetry of iteration graphs
- Tengxia Ju, Meiyun Wu, On iteration digraph and zero-divisor graph of the ring ${\mathbb{Z}}_{n}$
- Michal Křížek, Lawrence Somer, Sophie Germain little suns
- Uzma Ahmad, Syed Husnine, Characterization of power digraphs modulo $n$
- Uzma Ahmad, Husnine Syed, On the heights of power digraphs modulo $n$
- Yangjiang Wei, Jizhu Nan, Gaohua Tang, The cubic mapping graph for the ring of Gaussian integers modulo $n$
- Uzma Ahmad, Muqadas Moeen, The classification of finite groups by using iteration digraphs
- Michal Křížek, Lawrence Somer, 17 necessary and sufficient conditions for the primality of Fermat numbers

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