Elementary Number Theory

Topics

1. Basics: divisibility, unique factorization, congruences, primes, . . .
2. Usual Suspects: infinitude of primes, theorems of Fermat-Euler, two squares, quadratic reciprocity, . . .
3. Simple applications to cryptography, codes, primality tests, . . .
4. Rational Function Fields
5. Quadratic forms

Here's a link to last year's course.

Homework

Homework is always due one week after hand-out except when stated otherwise. Solutions will be posted after all students have turned their homework in.

07.02.06 homework 1; solutions.
14.02.06 homework 2; solutions.
23.02.06 homework 3 You can hand in problem 1 (RSA) later. Here's a windows executable of pari. If you type in ?, you'll get a list of chapters; ?4 lists e.g. the number theoretical functions, and ?gcd tells you what gcd does. You can find a more detailed manual at the pari homepage in Bordeaux.
14.03.06 homework 4; solutions.
11.04.06 homework 5; solutions.
27.04.06 homework 6; solutions.

Books

Here are a few books I recommend

Rose: A course in number theory
Hardy and Wright: An introduction to the theory of numbers
Ireland and Rosen: A classical introduction to modern number theory

It is sufficient to be familiar with the lecture notes on this page. This remark should not keep you from looking at other books, however.

Schedule

02.02.06 Monoids, unique factorization (sorry for hurrying through the last proof; I'll do it again next time).
07.02.06 residue classes
09.02.06 greatest common divisors, Euclidean algorithm
14.02.06 Euclidean algorithm, Pythagorean triples
16.02.06 Fermat's Theorem on the diophantine equation x⁴ + y⁴ = z⁴.
21.02.06 Fermat's Little Theorem; Z/pZ is a field.
23.02.06 RSA
28.02.06 Residue class rings; Pollard's p-1 method. here you can find the record factors found with this method.
02.03.06 NO CLASS: moved to 04.03.06
Sat. 04.03.06, 10:30-12:30 Midterm 1 in BZ08 (Computer Center Building). We will cover the material up to 21.02.06. Average: 67/100. Here are a few similar problems for those who need more practice.
07.03.06 Discussion of Midterm 1. Theorem of Euler-Fermat
09.03.06 Chinese Remainder Theorem
14.03.06 Chinese Remainder Theorem (applications); order of elements; Mersenne and Fermat numbers
16.03.06 primitive roots
21.03.06 primitive roots; quadratic reciprocity
23.03.06 NO CLASS: moved to 04.03
04.04.06 quadratic residues; Legendre symbol.
06.04.06 the quadratic reciprocity law
11.04.06 Quadratic reciprocity; Jacobi symbol.
13.04.06 discussion of homework; binary quadratic forms
18.04.06 homework; review; quadratic forms
20.04.06 Midterm 2, SBZ03, 5:40 - 7:40 pm (no class at 9:40). We will cover phi function, Euler-Fermat, RSA, Chinese Remainder Theorem, order of residue classes, primitive roots, quadratic reciprocity. Average: 71/100.
25.04.06 Reduction.
27.04.06 Class numbers
02.05.06 Class numbers
04.05.06 Quadratic number fields
09.05.06 Unique Factorization and class number 1
11.05.06 Composition
16.05.06 Composition; review
17.05.06 Final 12.15, SAZ04.
Here is the complete set of notes.
Here are a few problems for the last two chapters. Here are the problems for a make up midterm 2, along with partial solutions.
Final problems solutions. Average 73.5/100
We 18.05, 10:30 - 12:00 I'll be in my office.
Next week: Oberwolfach. Mathematicians at work.
Mo 29.05, 13:30 make up

Additional Lecture notes here (topics in n.th.), here (everything is in postscript), and here.