MATH 346
Number Theory, Spring 2009

Lectures:

MWF 3:00-3:50, Lillis Hall 185

Office hours:

Thursday 12:00-1:15 or by appointment

My office:

207 Deady, phone 3465635

Textbook:

Elementary Number Theory by Charles Vanden Eynden
2nd edition,

Homework:

Assignments will be posted on this web page. Homework will be collected on Fridays, at the beginning of class (with the exception of the first assignment that will be collected on Wednesday, April 8). Late homework will not be accepted. You are encouraged to collaborate on homeworks, however, writing up the solutions should be an individual work. Not all of the assigned homework problems will be graded. The lowest homework score will be dropped.

Midterm: will be held in class on May 4.

Final: June 8 at 3:15pm

Grading:

homework 30%
midterm 25%
final 45%

Schedule:

week 1 (March 30-April 3) Divisibility properties, Division with remainder, Euclidean algorithm (sections 1.1, 1.1, 1.2)

Assignment #1 (due April 8)

section 1.1: problems 4,6,8,18,24,26
section 1.2: problems 2,4,12,14,16,18,20,22,26
section 1.3: problems 1,7,10,11,18

Solutions to selected homework problems are on the Blackboard

week 2 (April 6-10) Congruences (sections 1.4, 1.5, 4.1)

Assignment #2 (due April 17)

section 1.4: problems 4, 10, 14, 22, 36, 37, 38
section 1.5: problems 10, 16, 28, 32, 42, 46
section 4.1: problems 4, 6, 14, 16

week 3 (April 13-17) Induction, algebra of congruence classes (sections 1.6, 2.1, 4.1, 4.2)

Assignment #3 (due April 24)

section 1.6: problems 21, 29, 41, 44, 60
section 2.1: problems 37, 38, 39
section 4.1: problems 26, 40, 42, 45, 49
section 4.2: problems 2, 20, 21

week 4 (April 20-24) Prime factorization, Theorems of Fermat and Euler (sections 2.1, 2.2, 4.2, 4.3)

Assignment #4 (due May 1)

section 2.2: problems 22, 24, 25
section 4.2: problems 24, 25, 44, 48
section 4.3: problems 16, 20, 34, 47, 48

Review problems for the midterm exam (covering sections 1.1-1.6, 4.1-4.2): 1.1.36, 1.2.36, 1.3.16, 1.3.28, 1.4.23, 1.4.35, 1.5.24, 1.5.36, 1.5.47, 1.6.35, 1.6.48, 4.1.17, 4.1.29, 4.2.11, 4.2.42, 4.2.49.

week 5 (April 27 - May 1) More examples with congruences

Assignment #5 (due May 11)

section 4.2: problem 51 (Hint: first, solve the following problem. Given a pair of distinct prime numbers p and q prove that there exists an integer n divisible by p^2 such that n+1 is divisible by q^2.)
section 4.3: problems 21, 24, 25, 39, 41, 44, 49

week 6 (May 4-8) Wilson's theorem, quadratic residues (sections 4.3, 5.3)

Assignment #6 (due May 15)

section 4.3: problems 26, 28
section 5.3: problems 2, 6, 8, 14, 19, 20, 43 (with proof).

week 7 (May 11-15) Quadratic reciprocity (section 5.4)

Assignment #7 (due May 22)

section 5.3: problems 45, 46 (Hint: prove by induction in n that if a is a quadratic residue modulo p then it is also a quadratic residue modulo p^n), 47
section 5.4: problems 14, 16, 24, 30, 37, 43.

week 8 (May 18-22) Primality testing (section 4.4)

Assignment #8 (due May 29)

section 4.4: problems 15, 19, 20, 27, 28, 29, 30
section 5.2: problems 29, 31, 32, 38, 40

week 9 (May 25-29) Primality testing, binomial coefficients

week 10 (June 1-5) RSA method, review

section 4.3: 29, 30, 31, 32, 33, 35, 36, 50;
section 4.4: 5, 6, 10, 11; section 4.5: 23, 24, 27, 28;
section 5.2: 30, 33, 34, 35; section 5.3: 38, 39, 40, 41;
section 5.4: 17, 19, 26, 28, 33, 49, 50.

1. How many entries in the 65th row of the Pascal triangle are odd?
2. Does there exists n such that all the entries of the n-th row of the Pascal triangle (except 1 at the beginning and the end) are divisible by 15? Explain.

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MATH 346 Number Theory, Spring 2009