470-2302/01 – Number Theory (TC)
| Gurantor department | Department of Applied Mathematics | Credits | 4 |
| Subject guarantor | RNDr. Pavel Jahoda, Ph.D. | Subject version guarantor | RNDr. Pavel Jahoda, Ph.D. |
| Study level | undergraduate or graduate | Requirement | Optional |
| Year | 3 | Semester | summer |
| | Study language | Czech |
| Year of introduction | 2012/2013 | Year of cancellation | |
| Intended for the faculties | FEI | Intended for study types | Bachelor |
Subject aims expressed by acquired skills and competences
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.
Teaching methods
Lectures
Tutorials
Summary
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.
Compulsory literature:
Compulsory literature is not required.
Recommended literature:
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.
Additional study materials
Way of continuous check of knowledge in the course of semester
Průběžná kontrola studia:
Studenti v průběhu semestru budou psát písemné testy. Za testy lze získat maximálně
30 bodů.
Podmínky udělení zápočtu:
K získání zápočtu je nutné získat minimálně 15 bodů.
E-learning
Basic materials are available on the educator's website: www.fei.vsb.cz/470/cs/osobni-stranky/jahoda/teorieCisel/
Other requirements
There are not defined other requirements for student.
Prerequisities
Subject has no prerequisities.
Co-requisities
Subject has no co-requisities.
Subject syllabus:
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
Conditions for subject completion
Occurrence in study plans
Occurrence in special blocks
Assessment of instruction