Schedule
This is a tentative schedule for the course. If necessary, I might change it.

2 Feb  4 Feb:
Introduction, Mathematical induction, Wellordering principle,
Division algorithm, Primes and Unique factorization.
Read: 1.2,1.3

9 Feb  11 Feb:
Euclid's algorithm, and Units and primes in a "System of numbers", and Euler's ϕ function:
Euler's theorem and Fermat's ``little'' theorem.
and modular arithmetic.
Read: 1.2 and 1.3 again, 2.1

16 Feb  18 Feb:
Mod m numbers. Chinese Remainder Theorem, Number
theory and publickey cryptography: the RSA algorithm.
Read: 2.2, 2.3, 2.5

23 Feb  25 Feb:
Solutions of polynomials modulo prime powers, Hensel's lemma.
Read: 2.6, 2.7

2 Mar  4 Mar:
Multiplicative order, Primitive roots, Quadratic residue.
Read: 2.8, 3.1

9 Mar  11 Mar:
Quadratic reciprocity, Midterm.
Read: 3.2

23 Mar  25 Mar:
Great integer function, Arithmetic functions, Convolution, and Möbius inversion.
Read: 4.1, 4.2, 4.3

30 Mar 1 Apr:
Geometry of numbers, Sum of two squares.
Read: 6.4

6 Apr  8 Apr:
Cyclotomic polynomials and its relation with primitive elements mod p, Beatty sequence and its generalization, Continued fractions.
Read: 7.1, 7.2

13 Apr  15 Apr:
Continued fractions (continued).
Read: 7.3, 7.4, 7.5, 7.6

20 Apr  22 Apr:
Continued fractions (continued). Solutions to Pell's equation.
Read: 7.6, 7.7, 7.8

27 Apr  29 Apr:
Quadratic fields, Failure of unique factorization.
Read: 9.5, 9.6, 9.7

Assignments.
 Due Feb. 11:
 Section 1.2: problems 47, 49, 51, 54;
 Section 1.3: problems 23, 36, 39;
 Due Feb. 18:
 Section 2.1: problems 10, 14, 20, 28.
 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?)
 How do you define a "prime" element in a "System of numbers"? (Any reasonable definition is fine as long as it passes the test for integers up to sign!) For which m's Z/mZ has only one prime element?

 Show that if p is prime and x^{2}1=0 in Z/pZ, then x is either 1 or 1. Is it true in Z/mZ where m is not prime?
 (Wilson's theorem) Show that (p1)!=1(mod p) if p is prime. Hint: Group any number from 1,..., p1 with its inverse modulo p.
 Due Feb. 25:
 Section 2.1: problems 29, 30, 36, 49.
 Section 2.3: problems 3, 7, 9, 13, 18, 21, 37, 41, 47.
 Due Mar. 4:
 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.
 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 Mar. 11:
 Section 2.7: Problem 12.
 Section 2.8: Problems 20, 24, 29.

 Let p be a prime number and k a positive integer. Show that Σ_{i∈ Z/pZ} i^{k} is divisible by p if k is not divisilble by p1 and it is congruent to 1 mod p if k is divisible by p1. (Hint: Use existence of primitive elements mod p.)
 Let f be a polynomial in (Z/pZ)[x]. If degree of f is at most p2, then p divides Σ_{i∈ Z/pZ} f(i)

 Let f be a polynomial in (Z/pZ)[x_{1},...,x_{n}] whose degree is less than n. If x_{1}^{i1}...x_{n}^{in} is a monomial of f(x)^{p1}, then at least one of the i_{j}'s is less than p1.
 1f(x)^{p1} is either 0 or 1 mod p. It is zero if x is not a solution of f≡0 mod p and it is 1 otherwise.
 Let f be a polynomial in (Z/pZ)[x_{1},...,x_{n}]. Show that if degree of f is less than n, then p dividesV_{f}(Z/pZ), where
V_{f}(Z/pZ):={a=(a_{1},...,a_{n})∈ (Z/pZ)^{n} f(a)≡ 0 (mod p)}.
 Determine all integers n>1 such that (2^{n}+1)/n^{2} is an integer.
 Due Mar. 25: No problem set.
 Due Apr. 1:
 Section 3.1, Problems 20, 21, 23.
 Section 3.2, Problems 15, 19, 20, 21, 24.
 Show that sum of quadratic residues mod p is divisible by p if p is an odd prime larger than 3.
 The last problem in the midtrem.
 Due Apr. 8: No problem set.
 Due Apr. 15
 Section 4.2, Problems 5, 10, 16, 19, 25.
 Section 4.3, Problems 15, 23, 25, 26, 28, 29.
 Section 7.1, Problem 3.
 Section 7.3, Problem 3.
 Section 7.4, Problem 1.
 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.)
 Due Apr. 22 No problem set.
 Due Apr. 29
 Section 7.4, Problem 6.
 Section 7.5, Problem 3.
 Section 7.6, Problems 3, 4.
 Section 7.7, Problems 2, 3.
 Section 7.8, Problem 5.
 Is there a function f from Q the set of rational numbers to {A,B} a set with two elements such that:
 f(1/x)≠f(x) for any nonzero x.
 f(x)≠ f(x) for any nonzero x.
 f(1x)≠ f(x) for any x.
Can this map be extended to the real numbers?
 Let ξ=(m+
√D
)/q and ξ'=(m
√D
)/q. If ξ has the following purely periodic simple continued fraction [a_0,...,a_n], then
1/ξ'=[a_n,a_(n1),...,a_0].
