Contact Information |

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

* 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.

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.
Szemer\'{e}di Regularity Lemma and Roth's theorem.

Week
Stuff
Lecture Notes
45
The origins of number theory in Euclid's
Elements (Fundamental Theorem of Arithmetic and the Infinitude of Primes).
PDF
46
Linear Diophantine equations and
Frobenius numbers.
PDF
47
Estimates for pi(x) from Euclid to Euler.
PDF
48
Squares (mod 4) and applications : (i)
Primes = 1 (mod 4) (ii) Fermat's theorem on the sums of two squares.
PDF
49
Dirichlet L-functions.
PDF
50
Bases in general :
Sidon sets, thin bases. Combinatorial and probabilistic number theory.
PDF
51
Thin bases (ctd.) : Chernoff's
inequality and Erdös theorem.
PDF
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

Exam 16/01/15
PDF
and solutions
PDF

Exam 27/08/15 PDF and solutions PDF

Exam yy/yy/15 PDF and solutions PDF

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