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