1. SylabUZ
2. Wydział Nauk Ścisłych i Przyrodniczych
3. semestr zimowy 2022/2023
4. Mathematics - pierwszego stopnia z tyt. licencjata
5. Number Theory

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Number Theory - opis przedmiotu

Cel przedmiotu

The aim is for the student to get the number theory course provided in the syllabus and to be able to apply it in practice in cryptography and computer science.

Wymagania wstępne

Zapisz zmiany

Linear Algebra 1 i 2

Zakres tematyczny

Lecture

1. Divisibility relation in the ring of integers (2 hours).
2. Least common multiple. Greatest common divisor and Euclid's algorithm, linear form for greatest common divisor, relationship between greatest common divisor and least common multiple. Coprime numbers. Fundamental theorem of arithmetic (2 hours).
3. Prime numbers. Canonical decomposition of a natural number into prime factors. Sieve of Eratosthenes. Goldbach's conjecture. Dirichlet's theorem. (6 hours).
4. Diophantine equations. (3 hours).
5. Congruences and their properties. Polynomial congruences and Lagrange's theorem. Wilson's theorem (3 hours).
6. Fermat's theorem on the decomposition of prime numbers of the form 4k+1 into the sum of two squares.
7. Chinese Remainder Theorem (3 hours)
8. Euler function and its properties. Euler's theorem and Fermat's little theorem (3 hours).
9. Arithmetic functions and their properties. Möbius and Liouvilla function (5 hours).
10. Legendre's symbol and his properties. Jacobi's symbol and its properties. Mersenne and Fermat numbers. Perfect numbers. Prime factors of Fermat numbers. Generalized sequences of Fermat numbers (5 hours).

Classes:

1. Proving properties of divisibility relations (2 hours).
2. Finding GCD and GCD for pairs of integers using the Euclidean algorithm, representing GCD using an appropriate linear form, and solving problems using the formula illustrating the relationship between GCD and GCD (3 hours).
3. Searching for prime numbers from a given interval using the Sieve of Eratosthenes, applying the canonical decomposition of a natural number, using calculated values of the p(x) function (3 hours).
4. Solving Diophantine equations using the matrix method. (3 hours).
5. Solving problems using the modulo congruence properties, proving certain congruences using Wilson's theorem, and applying the Chinese remainder theorem to problems (5 hours).
6. Calculating the values of the Euler function for natural numbers, calculating the remainder when dividing two natural numbers using Euler's theorem, and calculating the values of individual arithmetic functions (5 hours).
7. Solving congruences using Legendre's symbol (4 hours).
8. Solving problems with proofs using Fermat and Mersenne numbers (5 hours).

Metody kształcenia

Lectures: traditional lecture.

Exercises: joint solving of tasks related to the subject, exercises illustrating the application of theory, and discussion.

Efekty uczenia się i metody weryfikacji osiągania efektów uczenia się

Opis efektu	Symbole efektów	Metody weryfikacji	Forma zajęć

Warunki zaliczenia

1. Final test with tasks of varying difficulty, allowing for the assessment of whether the student has achieved the minimum learning outcomes.
2. Participation in lectures and a theoretical test at the last lecture.

The grade for the course consists of positive grades for exercises (60%) and lectures (40%).

Literatura podstawowa

1. L.E. Dickson, Introduction to the theory of numbers, New York 1957.
2. W. Sierpiński, Elementary Theory of Numbers, PWN, Warszawa 1987.

Literatura uzupełniająca

1. Ellina Grigorieva, Methods of Solving Number Theory Problems, Birkhäuse, 2018
2. Wacław Sierpiński, 250 Problems in Elementary Number Theory, New York, American Elsevier Pub. Co., 1970

Uwagi

Zmodyfikowane przez dr Sebastian Czerwiński (ostatnia modyfikacja: 07-02-2024 20:31)