# MTHT 435 Foundations of Number Theory

## Fall 2005

** Week 1** Divisibility, Euclid's Algorithm, Bezout's identity.

Read Chapter 1.

** Week 2** Linear Diophantine Equations, prime numbers, Fundametnal Theorem of
Arithmetic, there are infintely many primes

Read Chapter 2 (skiping 2.3)

** Week 3** there are infinitely many primes of the form 4n+3, finding all
pythagorean triples

Read 11.5 of text or Chapters 2,3 of Silverman.

** Week 4** rational solutions to x^2+y^2=1, congruences, solving linear
congruences.

Read Chapter 3.1, 3.2

** Week 5** simultaneous linear congruences, the Chineese Remainder Theorem
and applications, soving X^2=1 (mod N)

Read Chapter 3.3-3.5

** Week 6** ** Midterm 1 on Friday 9/30**, Fermat's Little Theorem

Read 4.1

** Week 7** Wilson's Theorem, Solving X^2=-1 (mod p), polynomials of degree d
have at most d roots mod p

Read 4.1

** Week 8** Hensel's Lemma, solving equations mod p^n, Euler's Theorem

Read 4.3,5.1,5.2

** Week 9** Multiplication formula for Euler's function, applications,
public key
cryptography, Rabin-Miller primality test

Read 5.3, Lecture Notes, chapters 16-19 of Silverman

** Week 10** Addition formula for Euler's function, primitive roots and
applications

Read 6.1,6.2, 6.6

** Week 11** Quadratic Residues, The Legendre Symbol,
Quadratic Reciprocity

Read 7.1--7.3,7.4

** Week 12** ** Midterm II Friday November 11** Quadratic Reciprocity

Read 7.4

**Week 13** Quadratic residues mod p^n,
Proof of Quadratic Reciprocity,
Quadratic Residues for arbitrary moduli, a generalization
of Hensel's Lemma

Read 7.4--7.6

**Week 14** Sums of two squares, THANKSGIVING

Read 10.1

**Week 15** The Equation X^4+Y^4=Z^4, Sums of four squares

Read 11.6, 11.7, 100.4