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Introduction to Number Theory
Introduction to Number Theory 0366-2140-02
Mikhail Borovoi
Spring 2013
Schedule:
Tuesday 14-16
Thursday 16-18
Syllabus:
The course is an introductory course in basic number theory.
It assumes very little background. The topics include
- The Euclidean algorithm, greatest common divisor, unique factorisation
into primes, linear Diophantine equations
- Continued fractions
- Congruences, the Chinese Remainder Theorem
- The multiplicative group of reduced residue classes modulo n,
Fermat's Little Theorem
- The Euclidean algorithm for polynomials
- Primitive roots
- Quadratic congruences, Legendre's symbol and quadratic reciprocity,
Jacobi's symbol
- Roots of polynomial congruences, Hensel's lemma
- The Prime Number Theorem (without proof) and its applications
- Primality testing
- Public Key Cryptography
- Pell's equation
- Arithmetic in the ring of Gaussian integers, sums of two squares,
Euclidean rings
- Pythagorean triples and Fermat's Last Theorem.
Bibliography
Any introductory book on number theory will be useful. For example, see:
- Elementary Number Theory, by D. Burton (available in Hebrew, published by the Open University).
- Elementary Number Theory: Primes, Congruences and Secrets, by William Stein (see online version).
- A more advanced text is "A Classical Introduction to Modern Number Theory"
by Ireland and Rosen.
Lectures in Hebrew on the Internet:
Number Theory
Homework: This is an important part of the course.
A student who does not do ALL the assignments will not be able
to pass the exam.
Because of budget constraints, only even-numbered assignments
(2,4,6,8,10,12) will be graded.
They should be turned in one week after they were given.
In order to be eligible to take the final exam,
at least 3 of the even-numbered assignments have to be
turned in in time.
10% of the final grade will be
determined from the homework scores,
which will be obtained as the average grade
of all the even-numbered assignments.
Odd-numbered assignments should not be turned in.
The solutions to them will be published on this page
two weeks after they were given.
Assignments:
#1,
#2,
#3,
#4,
#6,
#7
,
#8
,
#9
,
#10
,
#11,
#12
Homework #5 was not given.
Solutions:
#1,
#3,
#7,
#9,
#11,
#12,
Exam:
Note that at the exam in 2013 the students will have to answer all
the questions!
The formula for the final grade F in terms of the exam grade E
and the average score of the even-numbered home assignments A:
F=0.9*E+A.
(For example if E=70 and A=10 then F=73.)
Example of an exam
A useful Internet resource: Mathematics - Stack Exchange
Contact me at: borovoi@post.tau.ac.il. Please write in Subject: Number Theory (in English).
Course homepage:
http://www.tau.ac.il/~borovoi/courses/NumberTheory/NT.html