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Linear Diophantine equations and congruences
Kaňáková, Natálie ; Beran, Filip (advisor) ; Jančařík, Antonín (referee)
This bachelor's thesis summarizes and systematizes knowledge about congruences and linear Diophantine equations. This work is divided into two parts. The first part is dedicated to congruences. At first, it shows where we can find congruences in real life, congruence as a relation, its properties, and applications in calculating the last ciphers of large numbers, proofs of divisibility rules, or calculating the date of Easter. Afterward, we look into congruences containing unknowns - linear congruence equations. It looks into methods of solving linear congruences and illustrates them in exercises. The last topic of the first part is oriented on systems of linear congruences and the Chinese remainder theorem, both for non-coprime and coprime moduli, the algebraic version, applications in various types of problems, and modular representation of numbers. The second part of this thesis is dedicated to linear Diophantine equations - equations with integer solutions. It shows various methods of solving linear Diophantine equations with two, three, or more unknowns - the extended Euclidean algorithm, reduction method, substitution method, and others. This part also describes the relationship between linear congruences and linear Diophantine equations and the use of this relationship in solving both linear...
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Divisibility for talented students of secondary schools
Živčáková, Andrea ; Robová, Jarmila (advisor) ; Bečvář, Jindřich (referee)
This thesis is an educational text for high school students. It aims to teach them how to solve typical problems concerning divisibility found in mathematical correspondence seminars and mathematical olympiad. Basic notions from the theory of divisibility are recalled (e.g. prime numbers, divisors, multiples). Criteria of divisibility by 2 to 20 are introduced, as well as diophantine equations and practical applications of prime numbers in real life. One whole chapter is dedicated to problems and exercises. Powered by TCPDF (www.tcpdf.org)
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Elliptic Curves and Diophantine Equations
Klepáč, Adam ; Šťovíček, Jan (advisor) ; Shaul, Liran (referee)
Given an equation of the form f(x, y) = 0, where f is a polynomial in two variables with rational coefficients of degree lower or equal to three, we will study the properties of the set of its rational solutions. We will show that if f is irreducible and the degree of f is three, then the corresponding cubic curve is birationally equivalent to a special cubic curve, often called elliptic. Furthermore, we will define a group law on the set of rational points of an elliptic curve and finish with the proof of Nagell-Lutz theorem, which states that all rational points of finite order in such defined group have integral coordinates. 1
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Divisibility for talented students of secondary schools
Živčáková, Andrea ; Robová, Jarmila (advisor) ; Bečvář, Jindřich (referee)
This thesis is an educational text for high school students. It aims to teach them how to solve typical problems concerning divisibility found in mathematical correspondence seminars and mathematical olympiad. Basic notions from the theory of divisibility are recalled (e.g. prime numbers, divisors, multiples). Criteria of divisibility by 2 to 20 are introduced, as well as diophantine equations and practical applications of prime numbers in real life. One whole chapter is dedicated to problems and exercises. Powered by TCPDF (www.tcpdf.org)
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Heronian triangles
DOHNALOVÁ, Alice
Work includes choosen properties and problems pair with Heronian triangles. It's available like mathematical utility for work in special-interest mathematics on secondary school.
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