Contact Information |

Lecturer : Peter Hegarty, Rum MV:L3032, Tel.: (031) 7725371 eller (076) 6377873, hegarty@chalmers.se

* 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.

OBS! The following schedule is approximate and will be continuously updated.

Complexity of algorithms (Euclid's algorithm and integer factorisation).
First applications of FTA to non-linear Diophantine equations : Pythagorean triples and Fermat's Theorem.
First comments on the distribution of the primes.
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.
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.
Lagrange's theorem on sums of 4 squares.
Introduction to general additive number theory : sumsets.
A modern outlook : structure in dense random sets.
Van der Waerden's theorem.
The theorems of Roth, Szemeredi and Tao/Green (only stated).

Week
Stuff
Lecture Notes
44
The origins of number theory in Euclid's
Elements (Fundamental Theorem of Arithmetic and the Infinitude of Primes).
PDF
45
Linear Diophantine equations and
Frobenius numbers.
PDF
46
Estimates for pi(x) from Euclid to Euler.
PDF
47
Squares (mod 4) and applications : (i)
Primes = 1 (mod 4) (ii) Fermat's theorem on the sums of two squares.
PDF
48
Dirihlet L-functions.
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.
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

Exam 20/12/08
PDF
and solutions
PDF

Exam xx/04/09 PDF and solutions PDF

Exam xx/08/09 PDF and solutions PDF

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Peter Hegarty <hegarty@math.chalmers.se> Last modified: Mon Dec 08 20:10:00 CET 2008