We begin with the classical questions such as factorization, distribution of primes and finding integer solutions to algebraic equations. Second part of the course will deal with number fields and basics of algebraic number theory. Our ultimate goal will be to understand the quadratic reciprocity law and Dirichlet's theorem on arithmetic progressions in the context of class field theory. The last theme of the course is counting points on algebraic varieties over finite fields and the Weil conjectures. Though the statement of these conjectures is quite elementary, their proof took decades and had very high influence on the development of contemporary number theory.

The prerequisites are basic analysis, linear algebra and group theory. Lectures will be accompanied by numerous exercises intended to practice calculations with algebraic numbers,

In case one is motivated for further studies, we could recommend to look through the book of David A. Cox for higher reciprocity laws and to learn about the Birch and Swinnerton-Dyer conjecture. The latter concerns with the L-function of an elliptic curve which will be one of our last themes.

- Euclidean algorithm and Unique Factorization
- [IrelandRosen] chapter I
- Homework 1, due on October 7

- Calculations with Residue Classes and the Quadratic Reciprocity Law
- [IrelandRosen] chapters 3-5, [Koblitz1] chapters I and II, [Serre] § 3 of chapter I
- Homework 2, due on October 21 / solutions
- Homework 3, due on November 8 / solutions

- Some Modern Problems of Elementary Number Theory: Primality Tests, Factorization and Public Key Cryptosystems
- [Koblitz1] chapters III-V, [PanchishkinManin] chapter 2
#### Vigenere Cipher Attack

This link will be activated at 12:00 on Friday, November 18th and will contain a ciphertext enciphered with the Vigenere algorithm: The file will contain a sequence of lowercase characters with spaces and punctuation marks. Decipher it and send me either the key or the plaintext (first line would suffice) to the address vlasenko@maths.tcd.ie. First successful submission gets + 15% to the final mark, next 4 get +10%.

Here is some stuff that might be helpful:- a picture of Vigenere square
- an example, a sentence from "Naive.Super" enciphered with the key "erlendloe"
- vigenere.py, to run type "python" in a terminal and "import vigenere"

results

- Distribution of Primes, Riemann Hypothesis, Primes in Arithmetic Progressions
- [Lang1], [Serre] chapter VI
- Homework 4, due on December 6 (every part a), b), etc. in this homework is counted separately and marked with 1 pt)
#### On the distribution of primes:

- PARI/GP script approximating the Chebyshev psi function using zeros of the Riemann zeta function,

to run it on gstokes type "ssh gstokes", go to the folder where you stored the script, type "gp" to run PARI/GP and then "\r zetazeros.gp" to load the script

Unfortunately higher resolution graphics is disabled on gstokes, you will see more using your own installation of PARI - An animation showing how psi function is approximated by the sum of waves can be found here
- Tables of zeta zeros by Andrew Odlyzhko
- Lecture notes on prime numbers and Riemann hypothesis by Carl Erickson
- Wikipedia article containing the exact formula for the Chebyshev psi function from the lectures
- Wikipedia article on the Prime Number Theorem
- Chebyshev's elementary results can be found in [Chandrasekharan]

- PARI/GP script approximating the Chebyshev psi function using zeros of the Riemann zeta function,

- Polynomials and Algebraic Extensions; Algebraic Numbers
- [Lang2] chapters IV-VI, [IrelandRosen] chapters 12 and 13
- Notes of my introductory lecture on algebraic numbers explaining Minkowski's geometric approach; see also [PanchishkinManin] chapter 4 (in Part II)

- Diophantine Equations of Degree Two and Class Numbers
- [Davenport] chapters V and VI, [Cox]

- Irrationality and Diophantine Approximation
- [HardyWright] chapter IV, [PanchishkinManin] § 4 of chapter II

*p*-adic numbers- [Koblitz2] chapter I

- Cubic Diophantine Equations and Elliptic curves
- [Koblitz1] § 1 of chapter VI, [PanchishkinManin] § 3 of chapter 1 , [IrelandRosen] § 1 of chapter 18

- Finite fields
- [Koblitz2] § 1 of chapter III, [IrelandRosen] chapter 7

- Zeta function of an algebraic variety and Weil conjectures
- [Koblitz2] chapter V, [IrelandRosen] chapters 10, 11 and 18

- [IrelandRosen]
- K.Ireland, M.Rosen, A Classical Introduction to Modern Number Theory
- [Koblitz1]
- N. Koblitz, A course in Number Theory and Cryptography
- [Serre]
- J.-P. Serre, A Course in Arithmetic
- [PanchishkinManin]
- Yu.I.Manin, A.A.Panchishkin, Introduction to Number Theory, Part I: Problems and Tricks
- [Lang1]
- S. Lang, Math Talks for Undergraduates
- [Lang2]
- S. Lang, Algebra
- [Davenport]
- H.Davenport, The Higher arithmetic
- [Cox]
- D.A. Cox, Primes of the form x^2 + n y^2
- [HardyWright]
- G.H.Hardy, E.M.Wright, An Introduction to the Theory of Numbers
- [Koblitz2]
- N. Koblitz, p-adic Numbers, p-adic Analysis and Zeta-Functions
- [Chandrasekharan]
- K. Chandrasekharan, Introduction to Analytic Number Theory

PARI/GP calculator