Peter Hegarty, Rum MV:L3032, Tel.: (031) 7725371 eller (076) 6377873,
hegarty@chalmers.se
I will write my own lecture notes, 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 are 10 or so copies of this book in Cremona (as of October 19).
(HR) : G.H. Hardy, An introduction to the theory of numbers.
A new 2008 edition has been ordered for the library.
(N) : M.B. Nathanson, Elementary methods in number theory, Springer GTM
Series.
This is not in the library.
TBA.
There will be 22 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. These files will be
updated continuously as the course progresses. I cannot always guarantee that
the notes for a given lecture will be available in advance, but they will
certainly be available no more than a day or so afterwards.
OBS! The following schedule is approximate and will be continuously updated.
| Week |
Stuff |
Lecture Notes |
| 44 |
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 |
| 45 |
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 |
| 46 |
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 |
| 47 |
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 |
| 48 |
Dirihlet L-functions.
Lagrange's theorem on sums of 4 squares.
Introduction to general additive number theory : sumsets.
|
PDF |
| 49 |
Bases in general :
Sidon sets, thin bases. Combinatorial and probabilistic number theory.
|
PDF |
| 50 |
Thin bases (ctd.) : Chernoff's
inequality and Erdös theorem.
A modern outlook : structure in dense random sets.
Van der Waerden's theorem.
The theorems of Roth, Szemeredi and Tao/Green (only stated).
|
PDF |
| 20/12 |
Exam |
|
Homework 1 (due Nov. 14) and
solutions
Homework 2 (due Nov. 28) and
solutions
Homework 3 (due Dec. 12) and
solutions
| List of Examinable Proofs |
PDF
Exam 20/12/08
PDF
and solutions
PDF
Exam xx/04/09
PDF
and solutions
PDF
Exam xx/08/09
PDF
and solutions
PDF
300807 and solutions
170107 and solutions
170105 and solutions
160403 and solutions
180103 and solutions
220801 and solutions
070401 and solutions
270101 and solutions
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: Mon Dec 08 20:10:00 CET 2008