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Computational problems of elementary number theory
Widž, Jiří ; Porubský, Štefan (advisor) ; Staněk, Jakub (referee) ; Šimša, Jaromír (referee)
Title: Computational problems of elementary number theory Author: Mgr. Jiří Widž Department: Department of Mathematics Education Supervisor: Prof. RNDr. Štefan Porubský, DrSc. Institute of Computer Science of the Academy of Sciences of the Czech Republic Abstract: The central notion of this presented thesis is the concept of continued fractions. The origin of this concept as one of the oldest mathematical methods is shown here in its historical connections. The technique of continued fractions belongs to classical parts of mathematics. Although the general theory of continued fractions is manifold and layered considerably, in textbooks it is usually treated according to the intended purpose of its use. In the present text we have summarized the foundations of the general theory of convergence of continued fractions with an emphasis on the theory of simple continued fractions and their most common applications. We show several possibilities how the concept of continued fractions can be generalized to other structures such as the Gaussian integers or polynomial continued fractions. In the chapter devoted to matrix continued fractions we shall demonstrate the possibility how to extend it to non-commutative algebraic structures. We also show how the apparatus of continued fractions can be used to solve...
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Idempotents, Group Membership and their Applications
Porubský, Štefan
S.Schwarz in his paper [165] proved the existence of maximal subgroups in periodic semigroups and a decade later he brought [167] into play the maximal subsemigroups and thus he embodied the idempotents in the structural description of semigroups. Later in his papers he showed that a proper description of these structural elements can be used to (re)prove many useful and important results in algebra and number theory. The present paper gives a survey of selected results scattered throughout the literature where an semigroup approach based on tools like idempotent, maximal subgroup or maximal subsemigroup either led to a new insight into the substance of the known results or helped to discover new approach to solve problems. Special attention will be given to some disregarded historical connections between semigroup and ring theory.
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Semigroup Structure of Sets of Solutions to Equation X^s = X^m
Porubský, Štefan
Using an idempotent semigroup approach we describe the semigroup and group structure of the set of solutions to equation X^m = X^s in successive steps over a periodic commutative semigroup, over multiplicative semigroups of factor rings of residually finite commutative rings and finally over multiplicative semigroups of factor rings of residually finite commutative principal ideal domains. The analysis is done through the use of the maximal subsemigroups and groups corresponding to an idempotent of the corresponding semigroup and in the case of residually finite PID’s employing the available analysis of the Euler-Fermat Theorem as given in [11]. In particular the case when this set of solutions is a union of groups is handled. As a simple application we show a not yet noticed group structure of the set of solutions to x^n = x connected with the message space of RSA cryptosystems and Fermat pseudoprimes.
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Computational problems of elementary number theory
Widž, Jiří ; Porubský, Štefan (advisor) ; Staněk, Jakub (referee) ; Šimša, Jaromír (referee)
Title: Computational problems of elementary number theory Author: Mgr. Jiří Widž Department: Department of Mathematics Education Supervisor: Prof. RNDr. Štefan Porubský, DrSc. Institute of Computer Science of the Academy of Sciences of the Czech Republic Abstract: The central notion of this presented thesis is the concept of continued fractions. The origin of this concept as one of the oldest mathematical methods is shown here in its historical connections. The technique of continued fractions belongs to classical parts of mathematics. Although the general theory of continued fractions is manifold and layered considerably, in textbooks it is usually treated according to the intended purpose of its use. In the present text we have summarized the foundations of the general theory of convergence of continued fractions with an emphasis on the theory of simple continued fractions and their most common applications. We show several possibilities how the concept of continued fractions can be generalized to other structures such as the Gaussian integers or polynomial continued fractions. In the chapter devoted to matrix continued fractions we shall demonstrate the possibility how to extend it to non-commutative algebraic structures. We also show how the apparatus of continued fractions can be used to solve...
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