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 (, F_{α}). We characterize the mΘ set (, F_{α}) 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 (, F_{α}) 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 p^{k} 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 p^{k}. 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. |
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Copyright © The Author(s), 2023. Published by Science Publishing Group |
Modal Θ-valent Sets, Modal Θ-valent Congruences, Number Theory, p-valent Modal Residues
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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
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
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
@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} }
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 PY - 2023 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 SP - 11 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 -