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Main menu for Browse IS/STAG
Course info
KMA / QVKTC
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Course description
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Department/Unit / Abbreviation
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KMA
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QVKTC
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Academic Year
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2023/2024
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Academic Year
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2023/2024
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Title
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Selected topics in number theory
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Form of course completion
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Exam
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Form of course completion
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Exam
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Accredited / Credits
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Yes,
10
Cred.
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Type of completion
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Oral
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Type of completion
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Oral
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Time requirements
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Course credit prior to examination
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No
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Course credit prior to examination
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No
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Automatic acceptance of credit before examination
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No
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Included in study average
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NO
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Language of instruction
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-
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Occ/max
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Automatic acceptance of credit before examination
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No
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Summer semester
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0 / -
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0 / -
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0 / 0
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Included in study average
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NO
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Winter semester
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0 / 0
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0 / 0
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0 / 0
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Repeated registration
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NO
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Repeated registration
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NO
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Timetable
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Yes
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Semester taught
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Winter + Summer
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Semester taught
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Winter + Summer
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Minimum (B + C) students
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not determined
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Optional course |
Yes
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Optional course
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Yes
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Language of instruction
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-
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Internship duration
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0
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| No. of hours of on-premise lessons |
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Evaluation scale |
S|N |
| Periodicity |
every year
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| Specification periodicity |
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Fundamental theoretical course |
No
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| Fundamental course |
No
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| Fundamental theoretical course |
No
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| Evaluation scale |
S|N |
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Substituted course
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None
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Preclusive courses
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N/A
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Prerequisite courses
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N/A
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Informally recommended courses
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N/A
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Courses depending on this Course
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N/A
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Histogram of students' grades over the years:
Graphic PNG
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XLS
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Course objectives:
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Aims of the course: To profound and extend knowledge of the classical basic number theory and to show applications of this theory to other branches of mathematics.
Annotation:
Algebraic number theory, continued fractions, arithmetic functions, zeta(x) function, representation of numbers, asymptotic density.
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Requirements on student
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self-study, consultations
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Content
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- Algebraic number theory, algebraic numbers, algebraic integers, minimal polynomial, norm of the algebraic number, conjugates, high of the polynomial.
- Factorisation of algebraic numbers, primes, prime ideals, basics of Galois theory.
- Quadratic number fields, quadratic polynomials, determinant of polynomials.
- Continued fractions, basic properties, partial continued fraction, best approximation of numbers, simple continued fractions.
- Periodic continued fractions, continued fractions with terms of natural numbers, expression of irrational numbers by continued fractions.
- Arithmetical functions d(n), ?(n), ?(n) and ?(n), number of divisor sof the number, sum of divisor sof the number, asymptotic estimations of arithmetical functions.
- Mőbius function, Mőbius inversion formula, evaluation of Ramanujan's sums, perfekt numbers.
- The zeta(x) function, basic properties, Dirichlet series, multiplication of Dirichlet series, the analytical interpretion of Mőbius formula.
- The general problem of aditive arithmetic, partitions of numbers, Euler's theorems, the Roger-Ramanujan identities, Ramanujan's continued fractions.
- Square-free numbers, asymptotical behaviour,
- Representation of a number by the sum of two or four squares, the four-square theorem, representations of numbers by a larger number of squares.
- Representation of numbers by cubes and higher powers, biquadrates, the problem of Pruhet and Terry.
- Kronecker's theorem, state in one dimension and statement of the general theorem.
- Asymptotic density, upper and lower asymptotic density, basic properties, examples, e.g. asymptotic density for primes, squarefree numbers
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Activities
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Fields of study
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Guarantors and lecturers
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Literature
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Basic:
Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers. Oxford University Press, 1980. ISBN 978-0-19-921986-5.
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Basic:
G. Tenenbaum. Introduction to analytic and probabilistic number theory. Cambridge: Cambridge Univ. Press, 1995. ISBN 0521412617.
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Basic:
Marcus, Daniel A. Number fields. Springer-Verlag, New York-Heidelberg, 1977. ISBN 9780387902791.
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Recommended:
Everest, Graham; Ward, Thomas. An introduction to number theory. Graduate Texts in Mathematics, 232. Springer-Verlag, London, 2005. ISBN 1-85233-917-9.
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Recommended:
Newman, Donald J. Analytic number theory. Graduate Texts in Mathematics, 177. Springer-Verlag, New York, 1998. ISBN 0-387-98308-2.
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Recommended:
Bateman, Paul T.; Diamond, Harold G. Analytic number theory. An introductory course.. World Scientific Publishing Co. Pte. Ltd., Hackensack, New York, 2004. ISBN 9789812562272.
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Recommended:
Chan, Heng Huat. Analytic number theory for undergraduates. Monographs in Number Theory, 3. World Scientific Publishing Co. Pte. Ltd., Hackensack, 2009. ISBN 9789814271363.
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Recommended:
Nathanson, Melvyn B. Elementary methods in number theory. Graduate Texts in Mathematics, 195. Springer-Verlag, New York, 2000. ISBN 1475773927.
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Recommended:
Murty, M. Ram. Problems in analytic number theory. Graduate Texts in Mathematics, 206. Springer, New York, 2008. ISBN 1441924779.
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Recommended:
Guy, Richard K. Unsolved Problems in Number Theory. Springer-Verlag, 1981. ISBN 0-387-20860-7.
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On-line library catalogues
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