## Instructor: Dr. Nancy Eaton

E-mail me: eaton@math.uri.edu

Phone: 874-4439

Office: Rm 222, Tyler Hall

Office hours: Monday 2:00-5:00, or Tuesday by apt.

Visit my web page: WEB PAGESection 01-Kingston

Meets MWF: 9:00-9:50, Craw 221

Solutions to selected homework problems

Practice problems and Keys to exams

Portfolio

Students who require accommodations
and who have documentation from

Disability Services (874-2098)
should make arrangements with me as soon as possible.

Text: *Elementary Number Theory,* by Kenneth H. Rosen, 4th edition. There is a web
site provided by the author to supplement the text. www.awlonline.com/rosen

Course Content and Goals:

Number theory is the study
of properties of the Integers, { ... , -3, -2, -1, 0, 1, 2, 3, ... }
and the Natural Numbers, {1, 2, 3, 4, ...}. It is both elegant and beautiful.
It also has many interesting applications in computer science and cryptography.
These results can be understood and appreciated without much mathematical
sophistication. In this course, you learn to prove theorems as we
explore ideas and topics. We will cover the basics: sequences, divisiblity,
primes, factoring, greatest common divisors, congruences, Linear Diophantine
Equations and Möbius Inversion. Other topics will be covered
as time permits.

Motivation: Some examples from the exercises.

Example:

Show that if *a* is
an integer, then 3 divides *a ^{3}- a.*

Example:

The *lucky numbers *are
generated by the following sieving process: Start with the positive
integers. Begin the process by crossing out every second integer
in the list, starting your count with the integer 1. Other thatn
1, the smallest integer not crossed out is 3, so we continue by crossing
out every third integer left, starting the count with the integer 1.
The next integer left is 7, so we cross out every seventh integer left.
Continue this process, where at each stage we cross out every *k*th
integer left, where *k* is the smallest integer not crossed out, other
than 1, not yet used in the sieving process. The integers that remain
are the lucky numbers.

*Find all lucky numbers less
than 100.*

*Show that there are infinitely
many lucky numbers.*

Example:

A Chinese puzzle found in
the sixth century work of mathematician Chang Ch'iu-chien, called the
"hundred fowls" problem, asks: If a cock is worth five coins, a
hen three coins, and three chickens together are worth one coin, how many
cocks, hens, and chickens, totaling 100, can be bought for 100 coins?

Homework Assignments: (25% of your grade)

I assign many problems from
the book. You are expected to do them all. We will go over
some of the solutions in class. I will select from those assigned a
few to hand in. I will grade them and return them with written comments
on your work. See working together.
I will give you opportunities in class to put your work on the board.
Some homework problems will be given from the technology section.
For that you will need to use *Maple* or some other computer algebra
system. I will also give some assignments involving the web site:
www.awlonline.com/rosen.

Portfolio: (25% of your grade)

From those exercises collected
and returned to you, I will select 10 for you to rewrite and hand in
as your portfolio for the class. This is intended to be a collection
of well thought out proofs from the course. See working together.

Exams: (Tests 25%, Final 25%)

There will be two one-hour
tests on Wednesday, March 5th and Wednesday, April 23rd and a final exam
on Friday, May 9, 8:00 - 11:00 AM. The exams will be taken from
material that we covered in class, so take careful notes.

Sections Covered:

We will draw material from
the following chapters in the book.

CHAPTER& TOPIC

1 The Integers

3 Primes and
Greatest Common Divisors

4 Congruences

6 Some special
congruences

7 Multiplicative
Functions

Others as time permits.

Dates to remember:

**Exam 1: Wed. March 5**

**Exam 2: Wed. April 23**

**Portfolio due: Fri. May 2**

**Final Exam: Fri. May 9, 8:00-11:00
AM**

*Maple: * To complete the maple assignments, you must use a computer
that *Maple* installed on it. All computers in computer labs
on campus have *Maple* installed on them. The Math Dept. computer
lab in Tyler Hall, Rm 101 also has assistants to help you. Click
here to get the hours that it is open: Tyler 101. If
you have never used *Maple* before, you might want to do the introductory
worksheet: Download "Introduction
to Maple"(intro141.mws)

Homework assignments: Check this list frequently to get the latest homework assignments. It will be updated as we go. We will go over some of these in class, and I will ask that you hand some in.

To get started, for friday Jan
24, DO: 1.1, #8; 1.3, #10,29; 3.1, #22;
4.1, #15 These are also listed below.

Chapter |
Section |
Exercises |
Hand In |
Due |
# |

Chapt 1 | 1.1 | 3, 7, 8, 9, 13, 14, 15, 16 | 14 | 2/5 | 1 |

1.2 | 1, 2, 6, 7, 8, 13, 14, 20, 26, 27 | 8 | 2/5 | 1 | |

1.3 | 1, 3, 10, 29 | Handout: 1,2 | 2/5 | 1 | |

1.4 | 1-13, 16, 19, 21-23, 33, 34, 39, 40, 48 |
22 | 2/5 | 1 | |

Appendix A | Page 517 | 1-8 | 2a, 6a | 2/18 | 2 |

Chapt 3 | 3.1 | 1, 3, 6, 9, 10, 12, 13, 14, 17, 22, 24 Computations: 5, 6 |
10, 14, 24 Computations6 (maple) | 2/26 | 3 |

3.2 | 1, 3, 4, 5, 6, 7, 8, 15, 16, 22, 23 | 8, 16, 22 | 3/3 | 4 | |

Accepting late work and retries for | Last chance See * below |
3/17 | A | ||

3.3 | 1, 2, 3, 4 Computations: 2, using numbers in 1 |
2, 4 Computations2 (maple) | 3/21 | 5 | |

3.4 | 1, 5-13, 31-33, 35-40, 43, 48-50, 62-66, 69-71, Computations: 2 |
8, 10, 38, 40 Computations2 (maple) | 3/31 | 6 | |

3.5 | 1-4, 17 | 17 | 4/15 | 7 | |

3.6 | 1-6, 10-13, 21 | 2 | 4/15 | 7 | |

Chapt 4 | 4.1 | 1, 2, 4-15, 18, 22, 23, 26, 28 | 4, 8, 22 | 4/15 | 7 |

extra credit | extra credit | do only 5 | 4/9 | EC | |

4.2 | 1-3, 5-9 |
||||

4.3 | 1,2,3,4,7,8,9 |
2,4,8 |
|||

Chapt 6 | 6.1 | 1-7 |
all |
||

6.2 | |||||

6.3 | |||||

Chapt 7 | 7.1 | ||||

7.2 | |||||

7.3 |

* A for Amnesty. This is your last chance to hand in any of the problems that were due for homeworks 1, 2, 3, or 4. Also if you turned it in already and did not get full credit, you can redo the problem and hand it in again. PLEASE - clearly mark the Homework number, section number and problem number. Also, you must turn in the original so that I can see how many points I gave you the first time.

Solutions to selected homework problems and examples.

Selected exercises from chapter 1

Selected exercises from chapter 3

Practice problems and keys to Exams:

Practice for exam 1 Key to exam 1 Notes

Practice for Final Key to Practice problems for final

I will have office hours that day from 2 to 5pm
on May 2. Either hand it in that day, or stop by my office to ask
questions on the ones that you are not sure about and then hand it in on
May 5.

Give complete questions to each problem.
Write the solution as formally as you can. Use complete sentences.
Write it as if it would be published. In fact, I will publish selected
solutions on the web! Either type or write very neatly. If
you type, it is okay to type the sentences and leave spaces for writing
equations by hand. If you have access to an equation editor, that
would be best.

Selected problems for the portfolio: 1.2: 8; 1.4: 22;
Appendix 6a, 3.1: 10, 24; 3.2: 8; 3.4: 10, 38; 4.1:
22; Prove that the square root of 2 is irrational.

Put the problems in the above order and if you work with others, follow
the guide below.

Working
together:

I encourage you to work together
under the following

circumstances.

1. Begin the problem on your own,
do as much as you can.

2. Ask your professor to
help you with it.

3. Ask someone from the class
to explain the basic outline of a solution.

4. If working with someone
else, sometimes they will also ask you for your understanding of the basic
outline.

5. Take what you learned
and write out the solution on your own, using your own words.

6. Never copy word for word
from anyone else’s paper. In fact it is better not to look at anyone
else’s completed written solution or to show yours to anyone else.

7. If you still do not completely
understand the solution, you can ask your professor to look at what you
wrote and try to clear up any parts of the solution that are not completely
clear and accurate.

There are benefits to discussing the problems with your classmates. If you become stuck on a problem, fresh ideas from someone else might provide you with some new angles to try. In the academic community as well as in business and industry, people often work in teams. So, it is good to get some practice working in groups. Working with someone from class will help you to improve your math communication skills as well.

Also, to improve your verbal communication skills, you will be asked from time to time to present the solution of a homework problem to the class. You will be given extra points toward your homework grade for this.