# 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.

### 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.