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Topics of Number Theory

Topics of Number Theory

Code	10829
Year	1
Semester	S1
ECTS Credits	12
Workload	TP(45H)
Scientific area	Mathematics
Entry requirements	Degree in Mathematics, or other degrees with a strong mathematical component, recognized by the course committee.
Mode of delivery	Face-to-face.
Work placements	Not applicable.
Learning outcomes	This course unit aims to consolidate knowledge of Elementary Number Theory, its history, applications and links with the curricula of Mathematics in the 3rd Cycle of Basic Education and Secondary Education. At the end of the course unit the student should: - Know some basic results and techniques from the theory of partition of integers. - Understand and apply different representations of real numbers. - Be able to justify the irrationality of some notable numbers. - Be able to make connections between the course unit syllabus and curricula of Mathematics in the 3rd Cycle of Basic Education and Secondary Education.
Syllabus	I- Divisibility and primes: Euclidean algorithm; Unique Factorization Theorem; Fermat and Euler methods of factorization; sieve of Eratosthenes; Mersenne and Fermat primes; infinitude and density of prime numbers; perfect and friend numbers; Euler function; linear Diophantine equations. II- Congruences: mod p arithmetic; divisibility tests; "proof of nines"; perpetual calendar; control digits; Chinese remainder theorem; theorems of Fermat, Euler and Wilson; representation of numbers as sums of squares; primitive roots; Pythagorean triples; Fermat's last Theorem (n = 4). III- Integer partitions: Euler's identity; Euler pairs; Ferrers diagrams; Rogers-Ramanujan identities; method of generating functions. IV- Representations of real numbers: decimal base and period of the decimal expansion of a rational number; other number systems; representation of real numbers in continued fraction. V- Some irrational numbers: square root of 2, pi and "e"; approximation of irrationals by rationals.
Main Bibliography	1) Aigner, M., Ziegler, G., Proofs from THE BOOK, Third edition. Springer. 2004. 2) Andrews, G., Eriksson, K., Integer Partitions, Cambridge University Press. 2004. 3) Ore, O., Number Theory and its History, Dover. 1988. 4) Rosen, K., Elementary Number Theory and Its Applications, Third edition. Addison-Wesley Publishing Company. 1992. 5) Santos, J.O., Introdução à Teoria dos Números, IMPA, Colecção Matemática Universitária. 2000.
Language	Portuguese. Tutorial support is available in English.