Franz Lemmermeyer

# Elementary Number Theory

### Topics

• 1. Fundamentals: the construction of natural numbers, integers, and rational numbers from the Peano axioms.
• 2. Basics: divisibility, unique factorization, congruences, primes, . . .
• 3. Usual Suspects: infinitude of primes, theorems of Fermat-Euler, two squares, quadratic reciprocity, . . .
• 4. Simple applications to cryptography, codes, primality tests, . . .
• 5. p-adic numbers
• 6. Quadratic forms

### 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
• Frey: Elementare Zahlentheorie
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

• We 02.02.05: natural numbers
• Mo 07.02.05: integers and rational numbers
• We 09.02.05: Unique Factorization
• Mo 14.02.05: Euclidean Algorithm; Diophantine equations. Here is a proof of Lamé's Theorem that Fibonacci numbers produce the longest chains in the Euclidean algorithm.
• We 16.02.05: *Extra class* More diophantine problems
• Mo 21.02.05: FLT for exponent 4; Two squares
• We 23.02.05: no class
• Mo 28.02.05: Quadratic Reciprocity (updated 07/03)
• We 02.03.05: Quadratic Reciprocity
• Mo 07.03.05: Euler-Fermat
• We 09.03.05: *Extra class* Chebyshev
• Mo 14.03.05: the RSA cryptosystem
• We 16.03.05: Euclidean rings (updated 20/04)
• Mo 21.03.05: Gaussian Integers (updated 20/04). Check out Keith Conrad's notes on Gaussian integers. He also has notes on quadratic residues at the bottom of his page.
• We 23.03.05: *Extra class* : special cases of Dirichlet's theorem on primes in arithmetic progressions.
• Mo 28.03.05: Residue classes in Z[i]
• We 30.03.05: Midterm 1 (review sheet)
• Mo 04.04.05: The Rational Function Field (updated 20/04)
• We 06.04.05: no class
• Mo 11.04.05: spring break
• We 13.04.05: spring break
• Mo 18.04.05: Legendre symbols in Z[i] and Fp[X]
• We 20.04.05: Quadratic Reciprocity in Z[i]; primitive roots
• Mo 25.04.05: Quadratic Reciprocity in Z via Gauss sums; Quadratic Reciprocity in Fp[X]
• We 27.04.05: Valuations, real and p-adic numbers; check out the links to the articles by Goldstein, Gouvea and Madore on this page
• Mo 02.05.05: counterexamples to the local-global principle. I also mentioned a problem I cannot solve: let M and N be monoids of natural numbers, and assume that M and N have unique factorization. Does this imply that the intersection of M and N also has unique factorization?
• We 04.05.05: no class
• Mo 09.05.05: discussion of homework problems
• We 11.05.05: CHANGE OF SCHEDULE: midterm postponed, no class
• Th 12.05.05: 18:00 (SAZ 04) Midterm 2; you should be able to do arithmetic in the rings Z[i], Fp[X], and Zp (chapters 10-13). Here are a few review problems (UPDATED 11.05.05, 19:30). solutions (average: 63)
• Mo 13.05.05:
• We 15.05.05: review
• Midterm 2 make-up problems solutions
• Final (here are some test problems.
I've been asked about cryptography by some. I will probably offer an unofficial course on crypto and coding next semester.
If you want to read something on your own, see whether you can find a copy of Buchmann's book in one of the libraries around here. You may also look at lecture notes on the web, such as those listed here.