| Course Unit Code | 470-2302/01 |
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| Number of ECTS Credits Allocated | 4 ECTS credits |
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| Type of Course Unit * | Optional |
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| Level of Course Unit * | First Cycle |
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| Year of Study * | Third Year |
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| Semester when the Course Unit is delivered | Summer Semester |
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| Mode of Delivery | Face-to-face |
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| Language of Instruction | Czech |
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| Prerequisites and Co-Requisites | Course succeeds to compulsory courses of previous semester |
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| Name of Lecturer(s) | Personal ID | Name |
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| JAH02 | RNDr. Pavel Jahoda, Ph.D. |
| Summary |
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| We meet the applications of the results of number theory daily, maybe unwittingly. A variety of systems of identification numbers, such as the postal slips (USPS-The United States Postal Service), in the barcodes (UPC-Universal Product Codes) or books (ISBN-International Standard Book Number). Furthermore, the results of the theory of numbers used for generating random numbers. You shall also apply them in various areas. In addition to statistics find its place even in the theoretical physics-particle simulations. Probably the most important applications has number theory in cryptography, are based on it the extremly safe encryption methods, yet easily applicable in practice. In the subject of elementary number theory students should acquire basic knowledge of mathematical apparatus, which stands for the above applications. Then they can understand how these applications work in practice. |
| Learning Outcomes of the Course Unit |
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| After completing the course the student will know the selected definitions of basic concepts of elementary number theory and the relations between them, understand their importance, and will be able to use his knowledge to the solution of the fundamental tasks of the theory of numbers. They will also understand the importance of these concepts for the solution of the selected application tasks - primality testing and the RSA encryption algorithm. |
| Course Contents |
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Lectures:
Divisibility on N and Z, the greatest common divisor, Euclidean algorithm,
Canonical decomposition,
The set of prime numbers — basic knowledge of the layout to the axis,
Prime-counting function, Tschebyshev inequality, the prime number theorem and Bertrand's postulate,
Asymptotic density of sets,
Congruence relation on Z,
Linear congruences,
Operation on Zn,
Euler's totient function,
Euler-Fermat's last theorem,
Miller-Rabin primality test,
RSA algorithm.
Practices
Properties of the divisibility on N and Z, Euclid's algorithm,
Link of the canonical decomposition algorithm with the greatest common divisor and least common multiple,
Presence of the prime numbers in arithmetical sequences and g-adic expansions of numbers,
Eratosthenes sieve,
Determining the densities of sets, asymptotic density of the set of prime numbers, Properties of congruence relation,
Solving of linear congruences,
Z_p field, Wilson's theorem,
The value of the Euler's function,
Examples on Fermat's primality test and Carmichael's numbers,
Examples on the Miller-Rabin primality test,
Examples on RSA algorithm |
| Recommended or Required Reading |
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| Required Reading: |
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| Compulsory literature is not required. |
| Není vyžadována žádná povinná literatura. |
| Recommended Reading: |
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APOSTOL T.M.: Introduction to Analytic Number Theory, Springer, 1976.
HARDY G.H., WRIGHT E.M.: An Introduction to the Theory of Numbers, Oxford, Clarendon press, 1954.
J.E. POMMERSHEIM, T.K. MARKS, E.L. FLAPAN, Number theory, USA: Wiley, 2010.
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KOLIBIAR M., LEGÉŇ A., ŠALÁT T., ZNÁM Š.: Algebra a príbuzné disciplíny, Bratislava, Alfa, 1992.
Jahoda, P. Základy teorie čísel a jejích aplikací pro nematematiky, elektronická verze http://mi21.vsb.cz/modul/zaklady-teorie-cisel-jejich-aplikaci-pro-nematematiky
ŠALÁT T. A KOL. Algebra a teoretická aritmetika 2, Alfa, Bratislava, 1986.
ZNÁM, Š. Teória čísel, Alfa, Bratislava, 1977.
APOSTOL T.M.: Introduction to Analytic Number Theory, Springer, 1976.
HARDY G.H., WRIGHT E.M.: An Introduction to the Theory of Numbers, Oxford, Clarendon press, 1954.
J.E. POMMERSHEIM, T.K. MARKS, E.L. FLAPAN, Number theory, USA: Wiley, 2010.
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| Planned learning activities and teaching methods |
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| Lectures, Tutorials |
| Assesment methods and criteria |
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| Task Title | Task Type | Maximum Number of Points
(Act. for Subtasks) | Minimum Number of Points for Task Passing |
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| Exercises evaluation and Examination | Credit and Examination | 100 (100) | 51 |
| Exercises evaluation | Credit | 30 | 15 |
| Examination | Examination | 70 | 35 |