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Lecturer: Giorgos Kapetanakis (email@example.com) Schedule: Wednesday and Friday 15.00-17.00 (Α214) Office Hours: Wednesday and Friday 14.00-15.00 (Γ212 Zoom link - Password: 2RcQwW) Please contact me via email if you want to arrange a meeting. Grading: Final Exam.
- Ι. Αντωνιάδης και Α. Κοντογεώργης, Θεωρία αριθμών και εφαρμογές. Εκδόσεις Κάλλιπος, 2015.
- Μ. Παπαδημητράκης, Θεωρία Αριθμών: Πρόχειρες Σημειώσεις.
- Ν. Τζανάκης, Θεμελιώδης Θεωρία Αριθμών.
- Δ. Πουλάκης, Θεωρία αριθμών, εκδόσεις Ζήτη, 1997.
- T. Apostol, Εισαγωγή στην αναλυτική θεωρία των αριθμών, Gutenberg, 2005. (μετάφραση Α. Ζαχαρίου και Ε. Ζαχαρίου)
- K. Rosen, Elementary Number Theory and Its Applications [6th edition], Pearson, 2011.
- 10 February - 16 February
Divisibility, Euclidean division, greatest common divisor, Euclidean algorithm, least common multiple, prime numbers (definition, basic properties, Euclid's theorem), the fundamental theorem of arithmetic. (Paragraphs 2.1-2.7 of )
- 17 February - 23 February
We applied the fundamental theorem of arithmetic on divisibility, the greatest common divisor and the least common multiple. We defined arithmetic functions and the Dirichlet product and saw some of its basic properties. We saw some relevant examples and exercises. (Paragraphs 2.8 and 3.1 of )
- 24 February - 1 March
We defined (completely) additive and (completely) multiplicative functions. We then focused on multiplicative functions and proved some of their properties. We then focused on the Möbius function, we proved the Möbius inversion formula and we saw the basic properties of Euler'sfunction. We concluded with some remarks on perfect numbers. (Paragraphs 3.2, 3.4, 3.5 and 3.6 of )
- 2 March - 8 March
We saw some exercises on arithmetic functions and answered those of the 1st set.
- 9 March - 15 March
We defined and saw the basic properties of the integers modulo. We defined the complete and the reduced residue systems modulo . We proved Wilson's theorem, Euler's theorem and Fermat's little theorem. We defined the order of an integer modulo and saw a few relevant exercises. (Paragraphs 4.1-4.6 of )
- 16 March - 22 March
Coronavirus won Number Theory this week...
- 23 March - 29 March
Coronavirus won Number Theory this week too...
- 30 March - 5 April
We solved the exercises of the 2nd set.
- 6 April - 12 April
Linear congruences and systems of linear congruences. The Chinese Remainder Theorem. We saw a few relevant examples. (Paragraphs 4.7 and 4.8 of )
- 13 April - 19 April
We solved the exercises of the 3rd set and saw a few additional examples. We introduced polynomial congruences. (Paragraphs 5.1-5.3 of )
- 20 April - 26 April
- 27 April - 3 May
Polynomial congruences modulo a prime power. Quadratic residues and the Legendre symbol. The Legendre symbol of -1 and 2, The quadratic reciprocity law. (Paragraphs 5.3, 6.1 and 6.2 of )
- 4 May - 10 May
The Jacobi symbol and relevant examples. Further, we answered the 4th set and we saw a few additional exercises. (Paragraph 6.3 of )
- 11 May - 17 May
We showed that primitive roots moduloexist if and only if or , where is an odd prime. Also, we counted the number of these roots (if they exist). We saw the RSA cryptosystem. (Paragraph 5.4 of )
- 18 May - 24 May
Introduction to Diophantine equations. Linear Diophantine equations. Pythagorian triples, Fermat's last theorem. (Paragraphs 8.1-8.4 of )
- 25 May - 31 May
The equation, Legendre's theorem. We answered the 5th set. (Paragraph 8.5 of )
- 1 June - 7 June
We solved the exercises of the 6th set.