Elementary number theory

2022/2023
Programme:
Mathematics Education
Year:
3 year
Semester:
second
Kind:
mandatory
ECTS:
6
Language:
slovenian
Hours per week – 2. semester:
Lectures
3
Seminar
0
Tutorial
2
Lab
0
Prerequisites

Completed course Algebra 2.

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

J. Grasselli: Elementarna teorija števil, DMFA 2009.
H. Dörrie: 100 Great Problems of Elementary Mathematics : Their History and Solution, Dover Publications, New York, 1982.
K. H. Rosen: Elementary Number Theory and Its Applications, Addison-Wesley, Reading, London, Amsterdam, 2000.
J. J. Tattersall: Elementary Number Theory in Nine Chapters, 2nd edition, Cambridge Univ. Press, Cambridge, 2005.

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

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]