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
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
Dirichlet 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
51
The lectures previously scheduled for this week have now been moved to previous weeks, so there will be no new material this week.
19/12
Exam. 08:30 - 12:30.

Homework 1 (due Nov. 16) and solutions

Homework 2 (due Dec. 3) and solutions

Homework 3 (due Dec. 14) and solutions

Exam 19/12/12
PDF
and solutions
PDF

Exam 02/04/13 PDF and solutions PDF

Exam 26/08/13 PDF and solutions PDF

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Peter Hegarty <hegarty@math.chalmers.se> Last modified: Tue Aug 27 13:55:00 CET 2013