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Introduction to Number Theory

Introduction to Number Theory

Code	16613
Year	3
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: Divisibility; 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 Related figures; 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. 6) Santos, J.O., Introdução à Teoria dos Números, IMPA, Colecção Matemática Universitária. 2000.
Teaching Methodologies and Assessment Criteria	There will be three assessments: Mini-Test 1, worth 4 points, to be administered on October 8, 2025 Mini-Test 2, worth 4 points, to be administered on November 10, 2025 Final exam, worth 12 points, to be administered on December 17, 2025
Language	Portuguese. Tutorial support is available in English.