Math 4150-B, Spring 2018

Intro to Number Theory
Georgia Tech

Tentative Schedule:

Date Topics covered Sections of book to read before class
January 9 Prime numbers 3.1-3.2
January 11 GCD, Euclidean Algorithm, and the Fundamental Theorem of Arithmetic 3.3-3.5
January 16 Linear Diophantine equations 3.7
January 18 Georgia Tech closed due to inclement weather
January 23 Linear congruences 4.1-4.2
January 25 Chinese Remainder Theorem 4.3
January 30 Polynomial congruences, Wilson's theorem 4.4, 6.1
February 1 Fermat's Little Theorem, Psuedoprimes, Euler's theorem 6.2, 6.3
February 6 Euler phi-function and divisor functions 7.1, 7.2
February 8 Perfect numbers, Mersenne primes, and Möbius inversion 7.3, 7.4
February 13 Midterm 1
February 15 Cryptology 8.1, 8.4
February 20 Primitive roots 9.1
February 22 Existence of primitive roots 9.2, 9.3
February 27 Discrete logarithms, Primality tests 9.4, 9.5
March 1 Psuedorandom numbers 10.1
March 6 Quadratic residues and reciprocity 11.1-11.2
March 8 Jacobi symbol 11.3
March 13 Decimal fractions, irrational and transcendental numbers 12.1
March 15 Midterm 2
March 20 Spring Break
March 22 Spring Break
March 27 Continued fractions 12.2, 12.3
March 29 Pythagorean triples 13.1
April 3 Fermat's Last Theorem 13.2
April 5 Sums of squares 13.3
April 10 Pell's equation 13.4
April 12 Congruent numbers 13.5
April 17/19 Elliptic curves Notes
April 24 Review for final exam

Syllabus: Click here for the syllabus.

Homework:

Homework 1: Due Thursday, January 25, 2018:

3.3: 22, 24

3.4: 6 (parts a and b)

3.5: 6, 8, 42

3.7: 2, 8

Homework 2: Due Thursday, February 1, 2018:

4.1: 6, 8, 24, 30

4.2: 2, 6, 10,14, 18

4.3: 4, 12, 36

Homework 3: Due Thursday, February 22, 2018:

7.3: 10, 30

7.4: 10, 16, 20, 34

8.4: 2, 8, 14

Homework 4: Due Thursday, March 1, 2018

9.1: 2, 6, 8, 16

9.2: 4, 12, 16

9.3: 4, 6, 12

Homework 5: Due Thursday, March 8, 2018

9.4: 2, 6, 8, 10, 18 (note that in the second sentence, at the end of the first line, the "3" should be a "p", so that it should read "every integer not divisible by p")

9.5: 2, 4

10.1: 4, 8, 20

Homework 6: Due Thursday, April 5, 2018

12.1: 6 (see the bottom of page 473 for the definitions of pre-period and period lengths), 20 (it is ok to cite Theorem 12.4, although it would be useful to think about why this result is true in the relevant case explicitly using calculations similar to those we did in class. For example, if a number x repeats with period a, then (10^a-1)*x has a terminating expansion, and so 10^b*(10^a-1)*x is an integer for some b. If x=1/n, then what does this tell you about a?)

12.2: 2, 4 (To get rid of a last number in the continued fraction expansion of 1, recall the example we gave in class for 1/2, and also take a look at the last formula on page 484), 8

12.3: 4 (Recall the fact that convergents of simple continued fractions give the best rational approximations to irrational numbers).

12.4: 6 (The bar denotes a repeating continued fraction, as on the very bottom of page 503. Imitate the example of the golden ratio from class.)

13.1: 2, 8, 26

Homework 7: Due Thursday, April 12, 2018

13.2: 4, 8, 10, 20

13.3: 2, 4, 6, 10, 18

Exams and Sample Exams:

Sample Exam 1

Exam 1 Solutions

Sample Exam 2

Exam 2 Solutions

Sample Final Exam