Schedule
This is a tentative schedule for the course. If necessary, I might change it.
 11 Sep:
Introduction: Primes and factorization.
Read: 1.2,1.3

15 Sep  19 Sep:
Unique factorization, Euclid's algorithm, and Congruences.
Read: 1.2 and 1.3 again

22 Sep  26 Sep:
Units and primes in a "System of numbers", and Euler's ϕ function:
Euler's theorem and Fermat's ``little'' theorem.
and modular arithmetic.
Mod m numbers.
Read: 2.1

29 Sep  3 Oct:
Chinese Remainder Theorem, Number
theory and publickey cryptography: the RSA algorithm.
Read: 2.3, 2.5

6 Oct  10 Oct:
Solutions of polynomials modulo prime powers, Hensel's lemma.
Read: 2.6, 2.7

13 Oct  17 Oct:
Multiplicative order, Primitive roots.
Read: 2.8

7. 20 Oct  24 Oct:
pvaluation, Primitive roots (continued), Quadratic residue, Midterm.
Read: 2.8, 3.1

3 Nov  7 Nov:
Polynomials over Z/pZ (review), Quadratic reciprocity.
Read: 2.7, 3.2

9. 10 Nov  14 Nov:
Quadratic reciprocity (continued), Great integer function, Arithmetic functions.
Read: 3.2, 4.1, 4.2

17 Nov  21 Nov:
Convolution, and Möbius inversion.
Read: 4.3

25 Nov:
Geometry of numbers, Sum of two squares.
Read: 6.4

1 Dec  5 Dec:
Cyclotomic polynomials and its relation with primitive elements mod p, Beatty sequence and its generalization, Continued fractions.
Read: 7.1, 7.2

8 Dec  12 Dec:
Continued fractions (continued).
Read: 7.3, 7.4, 7.5, 7.6

Assignments.
 Due Sep. 25:
 Section 1.2, problems 47, 49, 51, 54;
 Section 1.3, problems 23, 36, 39;
 Due Oct. 2:
 Section 2.1, problems 10, 14, 20, 28, 44.
 Find all the Pythagorean triples, i.e. all triples of natural numbers (a,b,c) such that a^{2}+b^{2}=c^{2}. (Hint:
 Use lines passing through (0,1) with rational slope.
 Rational points on the circle x^{2}+y^{2}=1.
 Use extra parameters, and give formulas for a, b, and c in terms of those extra parameters, e.g. one can write all the integer solutions of 3a+2b=1 as a=2t+1 and b=3t1, where t can be any integer number (convince yourself why?))
 Describe all the primes in Z/mZ. In particular, say under what condition, there is only one prime, up to multiplication by a unit, in Z/mZ.
 Which prime numbers can be written as sum of two perfect squares? (Hint:
 Look at a^{2}+b^{2} modulo 4.
 Use Wilson's theorem.
 As most of you have already noticed the hard part of this problem is in Lemma 2.13 of your book. I expect you to think on this problem, yourself. After a while, you can read proof of this lemma, understand it, and then answer this problem. Again let me emphasis that you have to write down your solution without looking at your book, i.e. you have to recreat the argument.
)
 Let n be an integer with g.c.d.(n,10)=1. Show that the decimal expansion of 1/n is repeating. (Hint: Look at the following equalities.
 1/9 = 1/10 + 1/100 + 1/1000 + . . . = .11111 . . .
 1/99 = 1/100 + 1/10000 + 1/1000000 + . . . = .01010101 . . .
 1/999 = .001001001 . . .
 1/9999 = .000100010001 . . .
Use their generalization, coupled with another theorem, to get the desired result.)
 Due Oct. 9:
 Section 2.1, problems 29, 30, 36, 49.
 Section 2.3, problems 3, 7, 9, 13, 18, 21, 37, 41, 47.
 Due Oct. 16:
 Section 2.5: Problems 3, 4, 5.
 Section 2.6: Problems 2, 7.
 Section 2.7, Problem 10.
 Show that there is no positive integer n for which 2^{n} + 1 is divisible by
7.
 Determine all integers n>1 such that (2^{n}+1)/n^{2} is an integer.
 Let n>3 be an odd number. Show that there is p a prime number such that
 p  2^{ϕ(n)}1,
 p does not divide n.
(Hint: Take a look at: Probem 35, section 2.3, Problem 49, section 1.2, and Problem 43, section 1.3.)
 Due Nov. 6:
 Problems 7, 8, 9, 10 from the midterm.
 Section 2.8, Problems 20, 24, 29, 32.
 Due Nov. 14:
 Section 2.7, Problem 12.
 Section 2.8, Problems 33, 34, 35
 Section 3.1, Problems 11, 15, 21, 23, 24 (There is a misprint. It should be (a,m)=1 instead of (a,p)=1.)
 Show that sum of quadratic residues mod p is divisible by p if p is an odd prime larger than 3.
 Due Dec. 2: (Happy Thanksgiving) (Extended to Dec. 4:)
 Section 3.1, Problem 20.
 Section 3.2, Problems 1, 7, 13, 15, 19, 20, 21, 24.
 Section 4.1, Problems 9, 20, 22, 23, 37.
 Section 4.2, Problems 5, 10, 16, 19, 25.
 Section 4.3, Problems 15, 23, 25, 26, 28, 29.
 Let &Phi_{n}(x) be the n^{th} cyclotomic polynomial.
 a) Show that if p divides &Phi_{n}(x) for some integer x and (p,n)=1, then p is of the form nk+1.
 b) Use the first part to show that there are infinitely many primes of the form nk+1.
 Due Jan. 8: (Happy new year)
 a) Let p be prime larger than 3. For some integer x, p divides x^{2}+x+1 if and only if p is of the form 3k+1.
b) Show that x^{2}+xy+y^{2}=1 is an ellipse. Its axis are lines y=x and y=x, and find its area. (Hint: Rotate by angle 45 degree. So x=(X+Y)/Sqrt(2) and y=(YX)/Sqrt(2), and rewrite the equation.)
c) If p is a prime of the form 3k+1, then there are integers x and y such that p=x^{2}+xy+y^{2}. (Hint: Use parts a, b, and Minkowski's convex body.)
 Section 7.1, problem 3.
 Section 7.3, problem 3.
 Section 7.4, problem 1.
 Section 7.5, problems 3,4.
 Section 7.6, problems 3,4,5.
