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Interesting prime numbers and their properties
KRATOCHVÍL, Daniel
The aim of this bachelor thesis is to introduce special kinds of prime numbers to the reader. The thesis is divided into four parts. The first, introductory, one deals with the definition of prime numbers, provides some basic information about them, and offers one of the methods for finding the prime numbers. The following two parts focus on special kinds of prime numbers, the former of them contains those kinds of prime numbers which are named after famous mathematicians. The latter part is concern with prime numbers which are special for their unique attribute. In the closing part there are some exercises either with or without their solution.
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Polynomial time primality testing
Bednaříková, Alžběta ; Žemlička, Jan (advisor) ; Čech, Martin (referee)
The topic of my thesis is the testing of prime numbers in polynomial time. The text focuses on the specific algorithm published in 2002 by Manindra Agrawal, Neeraj Kayal and Nitin Saxena and it is known as the AKS primality test. In the introduction of this work, important properties and concepts essential for the text understanding are revised. Then the basic idea of the test is explained. The description of the algorithm itself continues. The aim of the work is to prove the Theorem on the correctness of the AKS test from a gradually built-up theory and to calculate the time complexity of the algorithm. Finally, it is proved that the calculated time complexity is polynomial.
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What we know about natural numbers
KOŠÁKOVÁ, Alena
The aim of my bachelor thesis is to introduce students, teachers, but also parensts and supporters of mathematics with natural numbers, especially prime numbers. In the first chapter we will explain the definition of prime numbers and their use in everyday life, the second chapter is focused on prime twins and the third on prime tiplets. This is followed by a chapter that contains various methods for calculating and finding primes. The fifth chapter is devoted to prime numbers, which are named after the eminent mathematicians who dealt with this topic.
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p-Adic numbers
Dubiel, Richard ; Šťovíček, Jan (advisor) ; Žemlička, Jan (referee)
Univerzita Karlova v Praze Matematicko-fyzikální fakulta BAKALÁŘSKÁ PRÁCE Richard Dubiel p-adická čísla Katedra algebry Vedoucí bakalářské práce: Mgr. Jan Šťovíček, Ph.D. Studijní program: Matematika Studijní obor: obecná matematika Abstract: This thesis deals with construction of the field of p-adic numbers as a completion of rational numbers field and introduces several important properties of this field. It will introduce concepts of an absolute value, metric, ultrametric and completion of field with respect to absolute value. Then we introduce a p-adic absolute value - one that measures "how much" is a number divsible by a prime number p. Then we construct the completion of the field of rational numbers with respect to this absolute value - field of p-adic numbers. We show, how can one represent these numbers. At last, we introduce two of the most important results of the theory of p-adic numbers - Hensel lemma and Hasse-Minkowski theorem.
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