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Introduction to Number Theory

Code 16610
Year 2
Semester S1
ECTS Credits 6
Workload TP(60H)
Scientific area Mathematics
Entry requirements don´t have
Learning outcomes 1st - Recognize and apply the fundamental properties of the divisibility relation in integers;
2nd - Apply Euclid's algorithm to determine the greatest common divisor of two integers;
3rd - Recognize prime numbers as well as their fundamental properties;
4th - Apply some primality tests;
5th - Solve linear Diophantine equations in two variables;
6th - Apply the fundamental properties of the congruence relation module m;
7th - Solve linear congruences and systems of linear congruences;
8th - Use the various methods studied to encrypt and decipher messages.
Syllabus Chapter I: Introduction:
Greatest common divisor and properties;
Euclid's algorithm;
Prime numbers;
Fundamental Theorem of Arithmetic;
Primality tests;
Diophantine equations.

Chapter II: Congruencies
Introduction to congruencies;
Linear Congruences;
Chinese Remainder Theorem;
Linear congruency systems;

Chapter III: Special congruencies;
Wilson's theorem;
Fermat's Little Theorem;
Euler's theorem;

Chapter IV: Quadratic Residuals
Jacobi symbol
Gauss's lemma
Legendre symbol
Law of quadratic reciprocity

Chapter V: Cryptography
Affin ciphers;
Cryptosystems based on prime numbers
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) Koshy T., "Elementary Number Theory with Applications", 2nd Edition, Harcourt, Academic Press, 2007
4) Ore, O., Number Theory and its History, Dover. 1988.
5) Rosen, K., Elementary Number Theory and Its Applications, 6th Edition. Addison-Wesley Publishing Company. 2018.
Teaching Methodologies and Assessment Criteria
Particular emphasis is placed on continuous assessment, which allows the student to progressively demonstrate the acquired skills throughout the semester. The semester will consist of three written examinations.
Language Portuguese. Tutorial support is available in English.
Last updated on: 2023-10-03

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