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The mΘ Quadratic Character in the mΘ Set

Received: 2 September 2022     Accepted: 28 September 2022     Published: 23 January 2023
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Abstract

The modal Θ-valent logic is a logic that contains all the thesis of the classical logical calculus and, besides allows to express notions of possibility, of necessity, and more others. The modal Θ-valent sets are the supports in term of the structure of the Θ-valent rings. A Θ chr (mΘ) is a structure which is rich at the same time of inheritance in the meaning of the romanian academician Gr. C. Moisil, as the algebraic model of a such logic. The set contains the set and the elements such that the support of x is not congruent to 0 modulo n. In this paper the purpose is to define on , p prime, a notion of quadratic residues and quadratic character which respects its structure of mΘs. Hoping that this approach will bring something of interest to the notion of quadratic residues. First of all, we construct the modal Θ-valent congruences of (, Fα). We characterize the mΘ set (, Fα) and we then give some arithmetical and intrinsic mΘ parameters of which lead us to the notion of factorial of m without n in , the mΘ quotient of (, Fα) modulo () and a complete system of mΘ residues modulo , . After that, we define a p-valent modal quadratic residue, p prime. We characterize some properties of p-valent modal quadratic character and p-valent modal quadratic residue of pk which establish the difference between the mΘ Euler’s theorem and the Euler’s theorem in the classical arithmetic. Later, we establish the theorem for determining the p-valent modal quadratic character of with respect to pk. This theorem is a non-classical version of Gauss’s lemma. Finally, we establish an example introducing the law of quadratic reciprocity of Gauss.

Published in Mathematics and Computer Science (Volume 8, Issue 1)
DOI 10.11648/j.mcs.20230801.12
Page(s) 11-18
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2023. Published by Science Publishing Group

Keywords

Modal Θ-valent Sets, Modal Θ-valent Congruences, Number Theory, p-valent Modal Residues

References
[1] P. Mooree, P. Stevenhagen, Prime divisors of Lucas sequences, Acta Arith. 82 (1997), 403 − 410.
[2] T. Ono, An introduction to algebraic number theory, New York, 1990.
[3] F. Nemenzo, H. Wada, An elementary proof of Gauss’ genus theorem, Proc. Japan Acad. Sci. 68 (1992), 94 − 95.
[4] F. Lemmermeyer, Reciprocity Laws: From Euler to Eisenstein, Springer, Berlin, 2000.
[5] C. F. Gauss, UntersuchungenÜber hohere Arithmetik (trans. H. Maser) , American Mathematical Society, 2006.
[6] Steve Wright, Quadratic Residues and Non-Residues, Lecture Notes in Mathematics, LNM, Volume 2171, 2016.
[7] FL Tiplea, S. Iftene, G Teseleanu, On the distribution of quadratic residues and non-residues modulo composite integers and applications to cryptography, Applied Mathematics and Computation 372, 124993, 2020- Elsevier.
[8] C. Monico, M. Elia, Note on an additive characterization of quadratic residues modulo p, Journal of Combinatorics, Information and System Sciences 31, 209-215, 2006.
[9] F. Ayissi Eteme, Logique et Algèbre de structures mathématiques modales Θ-valentes chrysippiennes, Edition Hermann, Paris, 2009.
[10] F. Ayissi Eteme, Anneau chrysippien Θ-valent, CRAS, Paris 298, série 1, 1984, pp. 1-4.
[11] FL Tiplea, A brief introduction of quadratic residuosity based cryptography, Math. Pures Appl, 2021.
[12] F. L. Tiplea, Efficient Generation of Roots of Power Residues modulo Powers of Two , 10(6), 908, March 2022.
[13] J. A. Tsimi and G. Pemha, A mΘ spectrum of Reed- Muller codes, Journal of Discrete Mathematical Sciences and Cryptography (JDMSC), 2021.
[14] J. A. Tsimi and G. Pemha, An algorithm of Decoding of mΘ Reed-Muller codes, Journal of Discrete Mathematical Sciences and Cryptography (JDMSC), 2021.
[15] J. A. Tsimi and G. Pemha, On the Generalized modal Θ-valent Reed-Muller codes, Journal of Information and Optimization Sciences (JIOS), 2021.
[16] F. Lemmermeyer, Hermite’s identity and the quadratic reciprocity law , Elem. Math. July 2022.
Cite This Article
  • APA Style

    Gabriel Cedric Pemha Binyam, Laurence Um Emilie, Yves Jonathan Ndje. (2023). The mΘ Quadratic Character in the mΘ Set . Mathematics and Computer Science, 8(1), 11-18. https://doi.org/10.11648/j.mcs.20230801.12

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    ACS Style

    Gabriel Cedric Pemha Binyam; Laurence Um Emilie; Yves Jonathan Ndje. The mΘ Quadratic Character in the mΘ Set . Math. Comput. Sci. 2023, 8(1), 11-18. doi: 10.11648/j.mcs.20230801.12

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    AMA Style

    Gabriel Cedric Pemha Binyam, Laurence Um Emilie, Yves Jonathan Ndje. The mΘ Quadratic Character in the mΘ Set . Math Comput Sci. 2023;8(1):11-18. doi: 10.11648/j.mcs.20230801.12

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  • @article{10.11648/j.mcs.20230801.12,
      author = {Gabriel Cedric Pemha Binyam and Laurence Um Emilie and Yves Jonathan Ndje},
      title = {The mΘ Quadratic Character in the mΘ Set },
      journal = {Mathematics and Computer Science},
      volume = {8},
      number = {1},
      pages = {11-18},
      doi = {10.11648/j.mcs.20230801.12},
      url = {https://doi.org/10.11648/j.mcs.20230801.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20230801.12},
      abstract = {The modal Θ-valent logic is a logic that contains all the thesis of the classical logical calculus and, besides allows to express notions of possibility, of necessity, and more others. The modal Θ-valent sets are the supports in term of the structure of the Θ-valent rings. A Θ chr (mΘ) is a structure which is rich at the same time of inheritance in the meaning of the romanian academician Gr. C. Moisil, as the algebraic model of a such logic. The set  contains the set  and the elements  such that the support of x is not congruent to 0 modulo n. In this paper the purpose is to define on , p prime, a notion of quadratic residues and quadratic character which respects its structure of mΘs. Hoping that this approach will bring something of interest to the notion of quadratic residues. First of all, we construct the modal Θ-valent congruences of (, Fα). We characterize the mΘ set (, Fα) and we then give some arithmetical and intrinsic mΘ parameters of  which lead us to the notion of factorial of m without n in , the mΘ quotient of (, Fα) modulo () and a complete system of mΘ residues modulo , . After that, we define a p-valent modal quadratic residue, p prime. We characterize some properties of p-valent modal quadratic character and p-valent modal quadratic residue of pk which establish the difference between the mΘ Euler’s theorem and the Euler’s theorem in the classical arithmetic. Later, we establish the theorem for determining the p-valent modal quadratic character of  with respect to pk. This theorem is a non-classical version of Gauss’s lemma. Finally, we establish an example introducing the law of quadratic reciprocity of Gauss.},
     year = {2023}
    }
    

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  • TY  - JOUR
    T1  - The mΘ Quadratic Character in the mΘ Set 
    AU  - Gabriel Cedric Pemha Binyam
    AU  - Laurence Um Emilie
    AU  - Yves Jonathan Ndje
    Y1  - 2023/01/23
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    N1  - https://doi.org/10.11648/j.mcs.20230801.12
    DO  - 10.11648/j.mcs.20230801.12
    T2  - Mathematics and Computer Science
    JF  - Mathematics and Computer Science
    JO  - Mathematics and Computer Science
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    EP  - 18
    PB  - Science Publishing Group
    SN  - 2575-6028
    UR  - https://doi.org/10.11648/j.mcs.20230801.12
    AB  - The modal Θ-valent logic is a logic that contains all the thesis of the classical logical calculus and, besides allows to express notions of possibility, of necessity, and more others. The modal Θ-valent sets are the supports in term of the structure of the Θ-valent rings. A Θ chr (mΘ) is a structure which is rich at the same time of inheritance in the meaning of the romanian academician Gr. C. Moisil, as the algebraic model of a such logic. The set  contains the set  and the elements  such that the support of x is not congruent to 0 modulo n. In this paper the purpose is to define on , p prime, a notion of quadratic residues and quadratic character which respects its structure of mΘs. Hoping that this approach will bring something of interest to the notion of quadratic residues. First of all, we construct the modal Θ-valent congruences of (, Fα). We characterize the mΘ set (, Fα) and we then give some arithmetical and intrinsic mΘ parameters of  which lead us to the notion of factorial of m without n in , the mΘ quotient of (, Fα) modulo () and a complete system of mΘ residues modulo , . After that, we define a p-valent modal quadratic residue, p prime. We characterize some properties of p-valent modal quadratic character and p-valent modal quadratic residue of pk which establish the difference between the mΘ Euler’s theorem and the Euler’s theorem in the classical arithmetic. Later, we establish the theorem for determining the p-valent modal quadratic character of  with respect to pk. This theorem is a non-classical version of Gauss’s lemma. Finally, we establish an example introducing the law of quadratic reciprocity of Gauss.
    VL  - 8
    IS  - 1
    ER  - 

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Author Information
  • Department of Mathematics and Computer Sciences, Faculty of Sciences, University of Douala, Douala, Cameroon

  • Department of Mathematics and Computer Sciences, Faculty of Sciences, University of Douala, Douala, Cameroon

  • Department of Mathematics and Computer Sciences, Faculty of Sciences, University of Douala, Douala, Cameroon

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