Peter Hegarty, Rum MV:L3032, Tel.: (031) 7725371,
hegarty@chalmers.se
I will write my own lecture notes and homeworks, and hand out photocopied extracts from various texts as needed, so there is no required course literature. There are many good number theory books out there, in case you want to have a text of your own. Here are some possibilities :
(NZM)
I. Niven, H. Zuckerman and H. Montgomery, An introduction to the theory of numbers (5th edition), Wiley 1991.
There might be some copies of this book in Cremona.
(HR) : G.H. Hardy, An introduction to the theory of numbers.
A new 2008 edition is in the library.
(N) : M.B. Nathanson, Elementary methods in number theory, Springer GTM
Series.
This is not in the library.
There will be 23 lectures and 3 homeworks . Examination will
be by means of a written exam at the end of the course. The exam will be
graded out of 100 points. A mxaimum of 15 bonus points can be obtained from
the homeworks (N.B.: these are NOT obligatory !). You will need a minimum of
50 points in total to pass the course.
As we proceed, completed material will be marked in green.
The files of lecture notes, together with the material handed out in
class, contain the detailed content of the course. Initially, I will work from my notes from the previous edition of the course in 2012. Some changes are anticipated, in particular in the latter part of the course dealing with additive and combinatorial number theory. Any editions will be made in real time, and be available no more than a day after the corresponding lecture.
OBS! The following schedule is approximate and will be continuously updated.
| Week |
Stuff |
Lecture Notes |
| 45 |
The origins of number theory in Euclid's
Elements (Fundamental Theorem of Arithmetic and the Infinitude of Primes). Complexity of algorithms (Euclid's algorithm and integer factorisation).
|
PDF |
| 46 |
Linear Diophantine equations and
Frobenius numbers. First applications of FTA to non-linear Diophantine equations : Pythagorean triples and Fermat's Theorem.
First comments on the distribution of the primes.
|
PDF |
| 47 |
Estimates for pi(x) from Euclid to Euler.
The Prime Number Theorem (PNT). The Riemann zeta function and heuristic
arguments for PNT. Chebyshev's theorem. Primes
in arithmetic progressions : Dirichlet's theorem.
Back to algebra : the
ring Z/nZ (Chinese Remainder Theorem) and the group (Z/nZ)*.
Euler's phi-function. The Fermat/Euler theorem and primality testing.
|
PDF
Supplement |
| 48 |
Squares (mod 4) and applications : (i)
Primes = 1 (mod 4) (ii) Fermat's theorem on the sums of two squares.
Sums of squares and other classical problems in additive number theory.
Quadratic residues in general :
Euler's criterion, Gauss lemma and Quadratic reciprocity.
Quadratic forms.
|
PDF
Supplement |
| 49 |
Dirichlet L-functions.
Lagrange's theorem on sums of 4 squares.
Introduction to general additive number theory : sumsets.
|
PDF
Supplement |
| 50 |
Bases in general :
Sidon sets, thin bases. Combinatorial and probabilistic number theory.
|
PDF
Supplement 1: Cauchy-Davenport theorem
Supplement 2: Probabilistic method |
| 51 |
Thin bases (ctd.) : Chernoff's
inequality and Erdös theorem.
A modern outlook : structure in dense random sets.
Van der Waerden's theorem.
Szemer\'{e}di Regularity Lemma and Roth's theorem.
|
PDF
Supplement |
| 2 |
The lectures previously scheduled for this week have now been moved to previous weeks, so there will be no new material this week.
|
|
| 16/1 |
Exam. 08:30 - 12:30. |
|
Homework 1 (due Nov. 24) and
solutions, here is Q.6(ii)
Homework 2 (due Dec. 12) and
solutions
Homework 3 (due Jan. 7) and
solutions, here is Q.11
| List of Examinable Proofs |
PDF
Exam 16/01/15
PDF
and solutions
PDF
Exam 27/08/15
PDF
and solutions
PDF
Exam yy/yy/15
PDF
and solutions
PDF
260813 and solutions
020413 and solutions
191212 and solutions
180811 and solutions
181210 and solutions
201208 and solutions
300807 and solutions
170107 and solutions
170105 and solutions
160403 and solutions
180103 and solutions
220801 and solutions
070401 and solutions
270101 and solutions
2012
2010
2008
2006
2004
2002
2000
Om du har kommentarer, påpekanden eller annat att säga
om kursen, tryck
här
Peter Hegarty
<hegarty@math.chalmers.se>
Last modified: Sat Aug 29 16:13:00 CET 2015