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• W4380490730 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 (<i>m</i>Θ) 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 <img width=20 height=10 src=http://article.sciencepublishinggroup.com/journal/247/2471199/image002.png /> contains the set <img width=10 height=10 src=http://article.sciencepublishinggroup.com/journal/247/2471199/image007.png /> and the elements <img width=25 height=10 src=http://article.sciencepublishinggroup.com/journal/247/2471199/image003.png /> such that the support of <i>x</i> is not congruent to 0 modulo <i>n</i>. In this paper the purpose is to define on <img width=40 height=10 src=http://article.sciencepublishinggroup.com/journal/247/2471199/image008.png />, <i>p</i> prime, a notion of quadratic residues and quadratic character which respects its structure of <i>m</i>Θs. 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 (<img width=20 height=10 src=http://article.sciencepublishinggroup.com/journal/247/2471199/image002.png />, <i>F_α</i>). We characterize the <i>m</i>Θ set (<img width=20 height=10 src=http://article.sciencepublishinggroup.com/journal/247/2471199/image002.png />, <i>F_α</i>) and we then give some arithmetical and intrinsic <i>m</i>Θ parameters of <img width=20 height=10 src=http://article.sciencepublishinggroup.com/journal/247/2471199/image002.png /> which lead us to the notion of factorial of <i>m</i> without <i>n</i> in <img width=20 height=10 src=http://article.sciencepublishinggroup.com/journal/247/2471199/image002.png />, the <i>m</i>Θ quotient of (<img width=20 height=10 src=http://article.sciencepublishinggroup.com/journal/247/2471199/image002.png />, <i>F_α</i>) modulo (<img width=30 height=10 src=http://article.sciencepublishinggroup.com/journal/247/2471199/image004.png />) and a complete system of <i>m</i>Θ residues modulo <img width=30 height=10 src=http://article.sciencepublishinggroup.com/journal/247/2471199/image004.png />, <img width=25 height=10 src=http://article.sciencepublishinggroup.com/journal/247/2471199/image005.png />. After that, we define a <i>p</i>-valent modal quadratic residue, <i>p</i> prime. We characterize some properties of <i>p</i>-valent modal quadratic character and <i>p</i>-valent modal quadratic residue of <i>p^k</i> which establish the difference between the <i>m</i>Θ 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 <img width=65 height=10 src=http://article.sciencepublishinggroup.com/journal/247/2471199/image006.png /> with respect to <i>p^k</i>. This theorem is a non-classical version of Gauss’s lemma. Finally, we establish an example introducing the law of quadratic reciprocity of Gauss." @default.
• W4380490730 created "2023-06-14" @default.
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• W4380490730 date "2023-01-23" @default.
• W4380490730 modified "2023-09-25" @default.
• W4380490730 title "The &lt;i&gt;m&lt;/i&gt;Θ Quadratic Character in the &lt;i&gt;m&lt;/i&gt;Θ Set &lt;img width=&quot;40&quot; height=&quot;20&quot; src=&quot;http://article.sciencepublishinggroup.com/journal/247/2471199/image002.png&quot; /&gt;" @default.
• W4380490730 doi "https://doi.org/10.11648/j.mcs.20230801.12" @default.
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