Formula for Number of Fixed Points of Digraph Arising from The Relation a^6 ≡ b (mod n) And Some Generalization

Main Article Content

Ratinan Boonklurb
Wipawee Kunanopparat

Abstract

For an integer gif.latex?n such that gif.latex?n&space;\geq&space;2, this article provides a formula for number of fixed points of the digraph gif.latex?\Gamma(n,&space;6) which has the vertex set  gif.latex?V&space;=&space;\{0,&space;1,&space;2,&space;\ldots,&space;n-1\} and directed edges gif.latex?(a,&space;b)&space;\in&space;E&space;\subseteq&space;V&space;\times&space;V  if and only if gif.latex?a^6&space;\equiv&space;b&space;\pmod{n} by using the knowledge about the factorization of the cyclotomic polynomial and extend the result to formula for number of fixed points of digraph gif.latex?\Gamma(n,&space;k) where gif.latex?k is an integer such that  gif.latex?k&space;\geq&space;4  and  gif.latex?k&space;-&space;1  is a prime number.

Article Details

How to Cite
Boonklurb, R., & Kunanopparat, W. . (2021). Formula for Number of Fixed Points of Digraph Arising from The Relation a^6 ≡ b (mod n) And Some Generalization. Mathematical Journal by The Mathematical Association of Thailand Under The Patronage of His Majesty The King, 66(703), 42–61. Retrieved from https://ph02.tci-thaijo.org/index.php/MJMATh/article/view/239949
Section
Research Article

References

รตินันท์ บุญเคลือบ และ ธัญพิชชา ยอดแก้ว. (2561). ไดกราฟที่เกิดจากความสัมพันธ์ a^4 ≡ b (mod n). วารสารคณิตศาสตร์ โดยสมาคมคณิตศาสตร์แห่งประเทศไทย ในพระบรมราชูปถัมภ์, 63 (695), น. 9 - 18.

Boonklurb, R. and Yodkeaw, T. (2018). Digraph Arising from The Relation a^4 ≡ b (mod n). Mathematical Journal by The Mathematical Association of Thailand under The Patronage of His Majesty the King, 63 (695), p. 9 - 18.

อัจฉรา หาญชูวงศ์. (2542). ทฤษฎีจำนวน. กรุงเทพมหานคร: โรงพิมพ์แห่งจุฬาลงกรณ์มหาวิทยาลัย.

Harnchoowong, A. (1999). Theory of Numbers. Bangkok: Chulalongkorn Unversity Printery.

Ju, T. and Wu, M. (2014). On Iteration Digraph and Zero-Divisor Graph of Ring Z_n. Czechoslovak Mathematical Journal, 64 (139), p. 611 - 628.

Kanoksing, P. (2017). Structure of the quotient ring of polynomials over integer modulo m with the nth cyclotomic polynomial. (Senior Project). Chulalongkorn University, Faculty of Science, Department of Mathematics and Computer Science.

Meemark, Y. (2016). Theory of Numbers. Retrieved from http://pioneer.netserv.chula.ac.th/~myotsana/MATH331NT.pdf.

Rosen, K. H. (1999). Discrete Mathematics and Its Applications (4th ed.). McGraw-Hill International Edition.

Skowronex-KaziÓw, J. (2009). Properties of Digraphs Connected with Some Congruence Relations. Czechoslovak Mathematical Journal, 59 (134), p. 39 - 49.

Somer, L. and Křížek, M. (2011). The Structure of Digraphs Associated with the Congruence x^k ≡ y (mod n). Czechoslovak Mathematical Journal, 61 (136), p. 337 - 358.

Szalay, L. (1992). A discrete iteration in number theory. BDTF Tud. Közl., 8, p. 71 - 91 (in Hungarian).