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Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

4 Issues per year


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ISSN
1898-9934
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In This Section
Volume 20, Issue 4 (Dec 2012)

Issues

Basic Properties of Primitive Root and Order Function

Na Ma
  • Qingdao University of Science and Technology, China
/ Xiquan Liang
  • Qingdao University of Science and Technology, China
Published Online: 2013-02-02 | DOI: https://doi.org/10.2478/v10037-012-0031-z

Summary

In this paper we defined the reduced residue system and proved its fundamental properties. Then we proved the basic properties of the order function. Finally, we defined the primitive root and proved its fundamental properties. Our work is based on [12], [8], and [11].

  • [1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.

  • [2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.

  • [3] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.

  • [4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.

  • [5] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.

  • [6] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.

  • [7] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.

  • [8] Zhang Dexin. Integer Theory. Science Publication, China, 1965.

  • [9] Yoshinori Fujisawa and Yasushi Fuwa. The Euler’s function. Formalized Mathematics, 6(4):549-551, 1997.

  • [10] Yoshinori Fujisawa, Yasushi Fuwa, and Hidetaka Shimizu. Public-key cryptography and Pepin’s test for the primality of Fermat numbers. Formalized Mathematics, 7(2):317-321, 1998.

  • [11] G.H. Hardy and E.M. Wright. An Introduction to the Theory of Numbers. Posts and Telecom Press, China, 2007.

  • [12] Hua Loo Keng. Introduction to Number Theory. Beijing Science Publication, China, 1957.

  • [13] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5):841-845, 1990.

  • [14] Artur Korniłowicz. Collective operations on number-membered sets. Formalized Mathematics, 17(2):99-115, 2009, doi: 10.2478/v10037-009-0011-0. [Crossref]

  • [15] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.

  • [16] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relative primes. Formalized Mathematics, 1(5):829-832, 1990.

  • [17] Xiquan Liang, Li Yan, and Junjie Zhao. Linear congruence relation and complete residue systems. Formalized Mathematics, 15(4):181-187, 2007, doi:10.2478/v10037-007-0022-7. [Crossref]

  • [18] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441-444, 1990.

  • [19] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4):341-347, 2003.

  • [20] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.

  • [21] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.

  • [22] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.

  • [23] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.

About the article

Authors thank Andrzej Trybulec and Yatsuka Nakamura for the help during writing this article


Published Online: 2013-02-02

Published in Print: 2012-12-01



Citation Information: Formalized Mathematics, ISSN (Online) 1898-9934, ISSN (Print) 1426-2630, DOI: https://doi.org/10.2478/v10037-012-0031-z. Export Citation

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