Homework #1. Due Wednesday, January 22nd, in class
Reading:
1. For this homework assignment: Chapter 1 and Section 2.1.
2. Before the class on Wed, Jan 22: Section 2.2-2.4.
Problems:
Problem 1: The Fibonacci numbers f1 , f2 , . . . are dened recursively b
Homework #1. Due Wednesday, January 23rd, in class
Reading:
1. For this homework assignment: Chapter 1 and Section 2.1. Note that
some material in Chapter 1 (like the general solution to a linear diophantine
equation) was not explicitly discussed in class
Homework #2. Due Wednesday, January 30th, in class
Reading:
1. For this homework assignment: Chapter 3 up to the end of Section 3.3.
2. For the next two classes: the rest of Chapter 3 and the beginning of
Chapter 4.
Problems:
1. Let n1 , . . . , nk and m
Homework #3. Due Wednesday, February 6th, in class
Reading:
1. For this homework assignment: Sections 3.5 and parts of Chapter 4.
2. For the next two classes: Sections 4.3, 4.1 and 5.1. Also review the
denition of the ring of congruence classes Zn (see Ch
Homework #7. Due Wednesday, March 20th
Nothing is due in writing this week, but you should do Problems 1 and 2 as
a preparation for next weeks classes (this is especially relevant for the class
on Wednesday). Also, Problems 3-6 below will likely appear in
Homework #8. Due Wednesday, March 27th
Reading:
1. For this homework assignment: Chapter 7. Make sure to read 7.2, 7.5 and
7.6 (in class we did not discuss 7.5, 7.6 and most of 7.2).
2. For the next two classes: Chapter 8.
Problems:
1. Let p be an odd pri
Homework #10. Due Wednesday, April 17th
Reading:
1. For this homework assignment: Chapter 10, Sections 10.1-10.3 + this
weeks class notes
2. For the next two classes: Pells equation and continued fractions (see
Problem 1 below)
Problems:
1. Read about Pel
Homework #9. Due Wednesday, April 3rd
Reading:
1. For this homework assignment: Chapter 8 and parts of Chapter 7.
2. For the next two classes: Chapter 9 (rough plan is to cover Sections
9.1-9.5)
Problems:
1. Recall that in Exercise 7.20 it was proved that
Math 5653. Number Theory. Spring 2013. First Midterm.
Wednesday, February 27th, 2-3:20pm
Directions: No books, notes, calculators, laptops, PDAs, cellphones, web
appliances, or similar aids are allowed. All work must be your individual
eorts.
Show all yo
Homework #11. Due on Tuesday, April 30th
Reading:
1. For this homework assignment: class notes on Pells equation and Chapter
11, Sections 11.1-11.4.
Some terminology and notations on continued fractions:
Let a0 , a1 , a2 , . . . be a nite or innite sequen
Homework #5. Due Wednesday, February 20th, in class
Reading:
1. For this homework assignment: Chapter 5 and Sections 6.1 - 6.2.
2. For the next two classes: Chapter 6.
Problems:
1. Let p be a prime. For a nonzero integer x, denote by ordp (x) the largest
Homework #4. Due Wednesday, February 13th, in class
Reading:
1. For this homework assignment: Chapter 4 and Section 5.1.
2. For the next two classes: Chapter 5 and Section 6.1.
Problems:
1. Let R be a commutative ring with 1. Prove that R , the set of uni
Homework #3. Solutions to selected problems.
1. Let p be a prime. As in class, for a nonzero integer x, denote by ordp (x)
the largest integer e s.t. pe divides x (if p x, we set ordp (x) = 0). We also
put ordp (0) = , so that we get a function ord : Z Z0
Homework #4. Due Thursday, February 13th, by 4pm
Reading:
1. For this homework assignment: Chapter 4 and Section 5.1.
2. For the next two classes: Chapter 5 and Section 6.1.
Problems:
1. Let R be a commutative ring with 1. Prove that R , the set of units
Homework #3. Due Thursday, February 6th, by 4pm
Reading:
1. For this homework assignment: Sections 3.5 and parts of Chapter 4.
2. For the next two classes: Sections 4.3, 4.1 and 5.1. Also review the
denition of the ring of congruence classes Zn (see Chapt
Homework #4. Solutions to selected problems
3. Let G be a nite group. The exponent of G, denoted by exp(G), is
the smallest positive integer m such that g m = e for all g G. Note that
g |G| = e for all g G by (a corollary of) Lagrange theorem, so we alway
Homework #2. Due Thursday, January 30th, by 4pm
Reading:
1. For this homework assignment: Chapter 3 up to the end of Section 3.3.
2. For the next two classes: the rest of Chapter 3 and the beginning of
Chapter 4.
Problems:
1. Let n1 , . . . , nk and m be
Homework #5. Due Wednesday, February 19th, in class
Reading:
1. For this homework assignment: Chapter 5.
2. For the next two classes: Chapter 6.
Problems:
1. Let n 2 be an even integer. Prove that for any a Z the congruence
x2 + 3x + a 0 mod n always has
Homework #6. Due Friday, March 7th, by 4pm
Reading:
1. For this homework assignment: Chapter 6.
2. For the next two classes: Chapter 6 and beginning of Chapter 7.
Problems:
1.
(a) Let G1 , . . . , Gk be nite groups. Prove that
exp(G1 . . . Gk ) = lcm(exp(
Homework #5. Solutions to selected problems
1. Let n 2 be an even integer. Prove that for any a Z the congruence
x2 + 3x + a 0 mod n always has an even number of reduced solutions
(possibly zero solutions).
Solution: Let f (x) = x2 + 3x + a. Below we deno
Homework #6. Due Wednesday, March 6th, in class
Reading:
1. For this homework assignment: Chapter 6.
2. For the next two classes: beginning of Chapter 7.
Problems:
1.
(a) Let G be a cyclic group of order n, and let m be a positive integer. Let
Pm (G) be t
Number Theory, Spring 2013. Midterm #2. Due Wednesday, April 10th
Directions: Provide complete arguments (do not skip steps). State clearly
and FULLY any result you are referring to. Partial credit for incorrect solutions, containing steps in the right di