Completed course Algebra 2.
Elementary number theory
2024/2025
Programme:
Mathematics, First Cycle
Year:
3 year
Semester:
second
Kind:
optional
Group:
B
ECTS:
6
Language:
slovenian
Course director:
Lecturer (contact person):
Hours per week – 2. semester:
Lectures
3
Seminar
0
Tutorial
2
Lab
0
Content (Syllabus outline)
Natural numbers and integers, fundamental theorem of arithmetic, multiplicative functions, Moebious inversion. Basic properties and distribution of primes.
Greatest common divisor, extended Euclidean algorithm. Finite and infinite continued fractions, best approximations, periodic continued fractions.
Congruences, Euler's function, Euler's theorem, Wilson's theorem. Encryption. Polynomial congruences, quadratic residues, Legendre symbol, quadratic reciprocity.
Diophantine equations: linear, quadratic (Pythagorean triples, Pell's equation). rational points on conics.
Sums of squares (sums of two, three or four squares). Lagrange's theorem. Integer binary quadratic forms: reduced forms, automorphisms, representations of numbers.
Readings
- H. Dörrie: 100 great problems of elementary mathematics : their history and solution, New York : Dover publications, cop. 1965.
- J. Grasselli: Elementarna teorija števil, Ljubljana : DMFA - založništvo, 2009.
- K. H. Rosen: Elementary number theory : and its applications, 5th ed. - Boston : Pearson, cop. 2005.
- J. J. Tattersall: Elementary number theory in nine chapters, Cambridge : Cambridge university, cop. 1999.
Objectives and competences
Students acquire the basic knowledge and skills in elementary number theory. Solving the elementary problems, students enhance their mathematical thinking and comprehension. The course by its content and methods of teaching deepens a prospective teacher's essential mathematical knowledge and skills.
Intended learning outcomes
Knowledge and understanding:
Knowledge and comprehension of essential concepts and definitions of elementary number theory and acquired ability to use these methods in elementary mathematical problems.
Learning and teaching methods
Lectures, tutorial sessions, individual consultations
Assessment
Course grade consists of two grades.
Exercise-based exam.
Theoretical knowledge exam.
Grades: 6-10 (pass), 5 (fail) (according to the Statute of UL).
Lecturer's references
Urban Jezernik:
JEZERNIK, Urban, SÁNCHEZ, Jonatan. Irrationality of generic quotient varieties via Bogomolov multipliers. International mathematics research notices, ISSN 1073-7928, vol. 2024, iss. 1, Jan. 2024, str. 284-330. [COBISS-SI-ID 143081475]
GARCÍA-RODRÍGUEZ, Javier, JAIKIN-ZAPIRAIN, Andrei, JEZERNIK, Urban. Units of group rings, the Bogomolov multiplier and the fake degree conjecture. Mathematical proceedings of the Cambridge Philosophical Society, ISSN 0305-0041, vol. 163, iss. 1, 2017, str. 115-123. [COBISS-SI-ID 18063449]
TRSTENJAK, Saša. Hassejevo načelo : magistrsko delo. Ljubljana: [S. Trstenjak], 2022, 78 str., ilustr. [COBISS-SI-ID 119343619]
Sašo Strle:
STRLE, Sašo. Bounds on genus and geometric intersections from cylindrical end moduli spaces. Journal of differential geometry, ISSN 0022-040X, 2003, vol. 65, no. 3, str. 469-511. [COBISS-SI-ID 13135193]
OWENS, Brendan, STRLE, Sašo. A characterization of the Z [sup] n [oplus] Z([delta]) lattice and definite nonunimodular intersection forms. American journal of mathematics, ISSN 0002-9327, 2012, vol. 134, no. 4, str. 891-913. [COBISS-SI-ID 16408153]
PREZELJ, Katja. Binarne kvadratne forme in cela števila : magistrsko delo. Ljubljana: [K. Prezelj], 2016. VI, 106 str., ilustr. [COBISS-SI-ID 17851481]
