Number Theory

Professor: Erika L.C. King

Email: eking@hws.edu

Office: Lansing 304

Phone: (315) 781-3355

Home Page

Office Hours: M: 1:30-3:30pm, W: 2:00-4:00pm, F: 10:30am-Noon and by appointment

Class Schedule: held TTh 1:30-2:55pm in Napier 202

Course Syllabus

Proof Writing and Presentation Tips

Course Grade Scale

What's Special About This Number? Website

**Review Session:** Thursday, May 7 from 11:00am until NOON in Napier 202. Attendance is optional and you are welcome to come for any portion of the
review session. Bring lots of questions.

**Colloquium/Mathematics Movie:** I will be showing the movie "Hard Problems" on Wednesday, May 6 at 1:00pm in Napier 201. Attendance and engaged
participation in the movie can count towards your colloquium requirement or extra credit.

**Office Hours:**

- Wednesday, May 6: 2:45pm-4:15pm
- Thursday, May 7: 2:30pm-4:00pm
- Friday, May 8: 10:30am-Noon
- Monday, May 11: 1:30pm-2:30pm
- By appointment

**In-Class Final Exam:** The in-class final exam is on Sunday, May 10th from
7:00pm until 10:00pm in Napier 202.

**Take-Home Final Exam Due:** The take-home final exam is due Monday, May 11 by 3:00pm. You may turn in the take-home final at the in-class final exam or
**in person** any time before 3:00pm on Monday.

**Thanks for an great class! Have a great summer! Keep in touch!**

**Reading Assignment for class Tuesday, May 5:**

- Review the material in Chapters 1-6 (pages 7-81), particularly the material from the presentations. Try exercises and questions from the project sections that were not included in the presentations. Bring thoughts and questions, and be ready to answer questions on your project section if others have questions. We will spend class going over exam 2, talking about end of the semester details, addressing questions you bring, working on some practice problems, and doing partner evaluations.

**Collected Homework (Due Tuesday, May 5 by 4:00pm):**

- Complete the problems on this handout.
- Note that there are no Notebook Problems due this week.
- You may also resubmit the Notebook problems that were due April 8 or April 22. Note that May 6 is the last day I will accept resubmissions. Remember that you must turn in your previous draft(s) with any resubmission. Please do not write on your previous submissions after they have been returned.

**Note that I will give you a free 24-hr extension for the assignments due this Tuesday. NO assignments will be accepted after 4:00 on Wednesday.**

**Reading Assignment for class Tuesday, April 28:**

- Write up a nice proof of Euler's Theorem and make sure what we discussed last Thursday makes sense. Then work on any exercises you did not yet complete in the sections on Fermat's Little Theorem and Euler's Theorem (pages 56-60).
- Prepare your projects. Review the grading rubrics I sent you so that you are keeping in mind what I will be looking for. We will see presentations on April 28 and 30. If you would like to make overheads or copies for the class, let me know if you need assistance (this is not necessary, and might work more for some presenations than others). If you would like to use some sort of computer presentation (like Beamer or Power Point), let me know way in advance so that I can try to figure out how to set things up (if possible) in our classroom, or book a different classroom. Have fun!
- Read the material to be presented in each class so that you are ready to ask questions and learn the material. There will be at least one question on the final and likely questions on your last assignment pertaining to information shared in these presentations. Thus before class this Tuesday you should read about Wilson's Theorem and about Public Key Cryptography, which are covered on pages 61-71. It is not expected that you work through exercises or proofs before class, but you should familiarize yourself with the material.

**Since your projects are due on Friday, there will be no homework assignment due on Wednesday of this
week. However, you are HIGHLY encouraged to resubmit Notebook problems that were due April 1 (this week is the last chance
for this one), April 8 or April 22. Recall that you may turn in your resubmissions on any day of the week up to one month after
they were originally due. Remember that you must turn in your previous draft(s) with any resubmission. Please do not write on your previous
submissions after they have been returned. Note that our last assignment will be due on Tuesday, May 5. **

**Reading Assignment for class Thursday, April 30:**

- Review the material covered by Mark and Hermione, and by Chris and Duncan. If you have any questions about it, ask them and/or me!
- You should read the material to be presented in class so that you are ready to ask questions and learn the material. There will be at least one question on the final pertaining to the information given in the student presentations. Thus for this Thursday you should read about Euler's phi-function and sums of divisors and about how Euler's phi-function is multiplicative, which encompasses pages 77-81. It is not expected that you necessarily work through exercises or proofs before class, but you should familiarize yourself with the material.
- In addition to Justine's presentation, we will also discuss material in the sections on Lagrange's Theorem and Primitive Roots (pages 73-76). Read these four pages, do the two exercises and prove Theorems 6.1 and 6.4.
- For some history, read pages 84-86 and 95-97. Much of this information is about mathematician Sophie Germain. As we tend not to hear about the great women mathematicians, this is a good opportunity to learn about one!

**Remember that your write up for your Project is due Friday, May 1 by 4:00pm.**

**Due to needing to cover another professor's class this week, my office hours Monday of this week are altered. They are:
Monday, April 20th, 3:15-4:45PM. If you
cannot make these times and need to see me, please email me and make an appointment.**

**Reading Assignment for class Tuesday, April 21:**

**Remember you should be prepared to present at least two proofs every class**, and should work through all the exercises, examples and questions in the reading. If you cannot solve a proof, move on to the next Theorem and try again later. You can, of course, always ask me and each other questions!- Come to class ready to claim which theorems you are ready to present.
- Read pages 56-60 of Chapter 4 (the section entitled "An alternative route to Fermat's Little Theorem" is optional). Prove Theorems 4.14, 4.15, 4.16, 4.17, 4.31 and 4.32. Remember to do the exercises and questions too! Please come and discuss them with me if you are having difficulty finishing them on your own.
- Work on preparing your projects. We will see presentations on April 28 and 30. If you would like to make overheads or copies for the class, let me know if you need assistance (this is not necessary, and might work more for some presenations than others). If you would like to use some sort of computer presentation (like Beamer or Power Point), let me know way in advance so that I can try to figure out how to set things up (if possible) in our classroom. Have fun!

**Collected Homework (Due Wednesday, April 22 by 4:00pm):**

- Complete the problems on this handout.
- Notebook Problems: This week's notebook problems are also on this handout. (Remember: (1) Turn these in on a separate piece of paper from your other homework, (2) notebook problems must be done on your own, and (3) you should give extra attention to form for these proofs.)
- You may also resubmit the Notebook problems that were due March 23 (this is the last chance for this one), April 1 or April 8. Note that you may turn in your resubmissions on any day of the week up to one month after they were originally due. Remember that you must turn in your previous draft(s) with any resubmission. Please do not write on your previous submissions after they have been returned.

**Reading Assignment for class Thursday, April 23:**

**Remember you should be prepared to present at least two proofs every class**, and should work through all the exercises, examples and questions in the reading. If you cannot solve a proof, move on to the next Theorem and try again later. You can, of course, always ask me and each other questions!- Come to class ready to claim which theorems you are ready to present.
- Reread pages 59-60 of Chapter 4. Go back to page 57 and prove Theorem 4.18 (it should be quick!). Then prove Theorem 4.32. Remember to do the exercises and questions too! Please come and discuss them with me if you are having difficulty finishing them on your own.
- Read pages 73-74 of Chapter 6, Lagrange's Theorem. Prove Theorem 6.1 (note that they forgot to define that polynomial as $f(x)$ in the first sentence!).
- Work on preparing your projects. We will see presentations on April 28 and 30. If you would like to make overheads or copies for the class, let me know if you need assistance (this is not necessary, and might work more for some presenations than others). If you would like to use some sort of computer presentation (like Beamer or Power Point), let me know way in advance so that I can try to figure out how to set things up (if possible) in our classroom. Have fun!

**Rememember our second exam will be Sunday, April 12th at 4:00pm in Napier 102.**

**Since we have an exam on Sunday, there will be no homework assignment due on Wednesday of this
week. This will be a great time to make sure you are ready to present proofs and exercises in class!
Note that you are also welcome to turn in Notebook resubmissions, though none of the remaining problems
have a final deadline of this week. Also you should begin work on your projects.**

**Reading Assignment for class Tuesday, April 14:**

**Remember you should be prepared to present at least two proofs every class**, and should work through all the exercises, examples and questions in the reading. If you cannot solve a proof, move on to the next Theorem and try again later. You can, of course, always ask me and each other questions!- Come to class ready to claim which theorems you are ready to present.
- Reread/read pages 49-52 of Chapter 3. Prove Theorems 3.27, 3.28 and 3.29. Read pages 53-55 of Chapter 4. Prove Theorems 4.6, 4.8 and 4.9. Remember to do the exercises and questions too! Please come and discuss them with me if you are having difficulty finishing them on your own.

**Due to needing to attend an event at my son's school, my office hours Friday of this week are altered. They are:
Friday, April 17, 1:30-3:00pm. If you
cannot make these times and need to see me, please email me and make an appointment.**

**Reading Assignment for class Thursday, April 16:**

- Remember you should be prepared to present at least two proofs and should work through all the exercises, examples and questions in the reading. If you cannot solve a proof, move on to the next Theorem and try again later. You can, of course, always ask me and each other questions!
- Come to class ready to claim which theorems you are ready to present.
- Spend some time working on your projects!
- Read pages 55-56 of Chapter 4. Prove Theorems 4.6, 4.8, 4.9, 4.10, 4.13 and 4.14. Remember to do the exercises and questions too! Please come and discuss them with me if you are having difficulty finishing them on your own.

**Rememember our second exam will be Sunday, April 12th at 4:00pm in Napier 102.**

**Homework Rewrite (Due Monday, April 6 at 3:00pm):**

- Rework problem 2 from your Week 9 assignment. If you did not earn credit for this problem, you should to do this assignment! Turn in your original problem set with your resubmission.

**Reading Assignment for class Tuesday, April 7:**

- Good work on Thursday! Let's see if Tuesday can be even better!
**Remember you should be prepared to present at least two proofs every class**, and should work through all the exercises, examples and questions in the reading. If you cannot solve a proof, move on to the next Theorem and try again later. You can, of course, always ask me and each other questions!- Come to class ready to claim which theorems you are ready to present.
- Reread/read pages 49-52 of Chapter 3, and start reading Chapter 4. Prove Theorems 3.20, 3.24, 3.27, 3.28 and 3.29. Remember to do the exercises and questions too! Please come and discuss them with me if you are having difficulty finishing them on your own.

**Collected Homework (Due Wednesday, April 8 by 4:00pm):**

- Complete the problems on this handout.
- Notebook Problems: This week's notebook problems are also on this handout. (Remember: (1) Turn these in on a separate piece of paper from your other homework, (2) notebook problems must be done on your own, and (3) you should give extra attention to form for these proofs.)
- You may also resubmit the Notebook problems that were due March 9 (this is the last chance for this one), March 23 or April 1. Note that you may turn in your resubmissions on any day of the week up to one month after they were originally due. Remember that you must turn in your previous draft(s) with any resubmission. Please do not write on your previous submissions after they have been returned.

**Reading Assignment for class Thursday, April 9:**

- Remember you should be prepared to present at least two proofs and should work through all the exercises, examples and questions in the reading. If you cannot solve a proof, move on to the next Theorem and try again later. You can, of course, always ask me and each other questions!
- Come to class ready to claim which theorems you are ready to present.
- Be sure to work on the proof of the third part of Theorem 3.24. Remember to try using the definition of divides sooner in the proof than we did on Tuesday.
- Reread/read pages 49-52 of Chapter 3, and start reading Chapter 4. Prove Theorems 3.24, 3.27, 3.28, 3.29 and 4.6. Remember to do the exercises and questions too! Please come and discuss them with me if you are having difficulty finishing them on your own.

**Quiz 4 will be on Thursday, April 2nd. It will cover material beginning on page 35 through what
we cover in Tuesday's class.**

**Reading Assignment for class Tuesday, March 24:**

**Remember you should be prepared to present at least two proofs every class**, and should work through all the exercises, examples and questions in the reading. If you cannot solve a proof, move on to the next Theorem and try again later. You can, of course, always ask me and each other questions!- Come to class ready to claim which theorems you are ready to present.
- Review our proofs to Question 3.7 and Corollary 3.9 that we did right at the end of class. Bring questions to office hours or class if there is anything unclear.
- Reread/read pages 45-49 of Chapter 3. Prove Theorems 3.14, 3.16, 3.17, and 3.20. Remember to do the exercises too! Please come and discuss them with me if you are having difficulty finishing them on your own.

**Collected Homework (Due Wednesday, April 1 (No Foolin'!) by 4:00pm):**

- Complete the problems on this handout.
- Notebook Problems: This week's notebook problems are also on this handout. (Remember: (1) Turn these in on a separate piece of paper from your other homework, (2) notebook problems must be done on your own, and (3) you should give extra attention to form for these proofs.)
- You may also resubmit the Notebook problems that were due February 23 (this is the last chance
for this one), March 9 or March 23. Note that you may turn
in your resubmissions on any day of the week up to one month after they were originally due
**(I am adding a little flexibility with the one month rule due to spring break)**. Remember that you must turn in your previous draft(s) with any resubmission. Please do not write on your previous submissions after they have been returned.

**Reading Assignment for class Thursday, April 2:**

- Remember that class will begin with a quiz. The topics we discussed in class on Tuesday are included in the material from which the quiz is taken. Be sure you ask me questions if it does not make sense!
- Remember you should be prepared to present at least two proofs and should work through all the exercises, examples and questions in the reading. If you cannot solve a proof, move on to the next Theorem and try again later. You can, of course, always ask me and each other questions!
- Come to class ready to claim which theorems you are ready to present.
- After the quiz, Mark, Duncan and Hermione will present 3.16 and Justine and Chris will present 3.17. Be sure to meet before Thursday's class so that you are ready to do this. Please ask me questions if you have them!
- Reread/read pages 45-49 of Chapter 3. Prove Theorems 3.16, 3.17, 3.19, 3.20 and 3.24. Remember to do the exercises and questions too! Please come and discuss them with me if you are having difficulty finishing them on your own.

**Have a great spring break!!!**

**Collected Homework (Due Monday, March 23 by 4:00pm):**

- Complete the problems on this handout.
- You may also resubmit the Notebook problems that were due February 16 (this is the last chance
for this one), February 23 or March 9. Note that you may turn
in your resubmissions on any day of the week up to one month after they were originally due
**(I am adding a little flexibility with the one month rule due to spring break)**. Remember that you must turn in your previous draft(s) with any resubmission. Please do not write on your previous submissions after they have been returned.

**Reading Assignment for class Tuesday, March 24:**

- Come to class ready to claim which theorems you are ready to present.
- Read/reread pages 35-42 of Chapter 2. Note that Exercise 2.40 asks you to do some research on the internet. Bring some ideas about arithmetic progressions to class and be ready to discuss them. Be sure you have worked through all the exercises.
- Read pages 43-47 of Chapter 3. Now we return to congruences! Don't forget your results from Chapter 1. Prove Theorems 3.8 and 3.14. Be sure to work through all the Exercises and Questions as well.

**Reading Assignment for class Thursday, March 26:**

- Come to class ready to claim which theorems you are ready to present.
- If you have an alternate proof of Question 3.4 to the one that Justine presented, put it on the board before class begins. I would especially be interested in seeing if someone figured out how to apply the suggestions in the text in the paragraph before 3.4.
- Reread/read pages 44-48 of Chapter 3. Prove Theorems 3.8 and 3.14, and Corollary 3.9. Be sure to work through all the Exercises and Questions as well.

**EXAM BONUS (Due Friday, March 27 at 3:00pm):**

- Rework your exam for up to half points back. You may also do the bonus problem. If you choose to do this assignment (and you
**should**!!!) you must (1) complete it entirely on your own except for your book, class notes and discussions with me, (2) give extra attention to form as this is bonus and you do not have a time crunch, and (3) turn in your original exam with your resubmission. Note that you do not have to rewrite the entire exam in order to submit the bonus; you need only rewrite those problems for which you wish to earn back points.

**Quiz 3 will be on Thursday, March 12th. It will cover material beginning on page 27 through what
we cover in Tuesday's class.**

**Collected Homework (Due Monday, March 9 by 4:00pm):**

- Complete the problems on this handout. Note that more was added on Friday!
- You may also resubmit the Notebook problems that were due February 9 (this is the last chance for this one), February 16 or February 23. Note that you may turn in your resubmissions on any day of the week up to one month after they were originally due. Remember that you must turn in your previous draft(s) with any resubmission. Please do not write on your previous submissions after they have been returned.

**Reading Assignment for class Tuesday, March 10:**

- Come to class ready to claim which theorems you are ready to present.
- Meet with your group to finalize your proof for Theorem 2.34. Each group should start putting your proof on the board as soon as you arrive to class.
- Keep experimenting with evaluating $\pi(n)$ for larger numbers. I will be interested in hearing about your progress with this.
- Read/reread pages 35-42 of Chapter 2. Prove or revise your proofs to Theorems 2.35 (there are several ways to do this and I would love to see more than one from the class!), 2.37 and 2.38.
- The MAA's NumberADay website seems to have stopped being updated, unfortunately. You can check out the history of past postings, however, to see some interesting facts and properties of numbers. Do go to the What's Special About This Number? Website! Choose three of your favorite numbers and see what you can find out about them! Be ready to share with the class. Note that these websites have been posted at the top of the page here.

**Reading Assignment for class Thursday, March 12:**

- Come to class ready to claim which theorems you are ready to present.
- Review the notes we took on proving Theorem 2.38 and write a nice version of the proof to share. If you want to claim it, you can start putting it on the board before class starts, but you must be sure it can hide behind the screen during the quiz, so put it front and center!
- Read/reread pages 35-42 of Chapter 2. Note that Exercise 2.40 asks you to do some research on the internet. Bring some ideas about arithmetic progressions to class and be ready to discuss them. Be sure you have worked through all the exercises.
- Start reading Chapter 3. Prepare Exercise 3.1.
- *** We didn't talk about our favorite numbers and the facts you discovered about them last time, so be sure you have done this: The MAA's NumberADay website seems to have stopped being updated, unfortunately. You can check out the history of past postings, however, to see some interesting facts and properties of numbers. Do go to the What's Special About This Number? Website! Choose three of your favorite numbers and see what you can find out about them! Be ready to share with the class. Note that these websites have been posted at the top of the page here.

**Rememember our first exam will be Sunday, March 1st at 4:00pm in Napier 202.**

**Since we have an exam on Sunday afternoon, you do not have a collected assignment due on Monday.
Concentrate on preparing for class on Tuesday. Also remember that you can resubmit Notebook problems. The
last day I will accept a resubmission for the problem originally due on February 2 is Tuesday.**

**Due to needing to cover another professor's class this week, my office hours Wednesday of this week are altered. They are:
Wednesday, March 4th, 3:15-4:15PM. If you
cannot make these times and need to see me, please email me and make an appointment.**

**Reading Assignment for class Tuesday, March 3:**

- Remember you should be prepared to present at least one proof in class and should work through all the exercises in the reading. If you cannot solve a proof, move on to the next Theorem and try again later. You can, of course, always ask me and each other questions!
- Come to class ready to claim which theorems you are ready to present.
- Prepare a proof by induction of Lemma 2.8. We did the base cases in class, but it would be great to see the whole thing. Feel free to start putting it on the board before class!
- Keep experimenting with evaluating $\pi(n)$ for larger numbers. I will be interested in hearing about your progress with this.
- Reread pages 31-33 of Chapter 2. Prove or revise your proofs to Theorems 2.9, 2.18, 2.19 and 2.20.
- Remember that a reading assignment includes working through all exercises, examples and QUESTIONS in the reading as well proving/solving the specified Theorems and Lemmas so that you are ready to ask questions about, discuss and present the material.

**Reading Assignment for class Thursday, March 5:**

- Come to class ready to claim which theorems you are ready to present.
- Keep experimenting with evaluating $\pi(n)$ for larger numbers. I will be interested in hearing about your progress with this.
- Reread pages 31-33 of Chapter 2. Prove or revise your proofs to Theorems 2.18, 2.19, 2.20, and 2.32-2.34.

**Rememember our first exam will be Sunday, March 1st at 4:00pm in Napier 202.**

**Collected Homework (Due Monday, February 23 by 4:00pm):**

- Complete the problems on this handout.
- You may also resubmit the Notebook problems that were due February 2, February 9 or February 16. Note that you may turn in your resubmissions on any day of the week up to one month after they were originally due. Remember that you must turn in your previous draft(s) with any resubmission. Please do not write on your previous submissions after they have been returned.

**Reading Assignment for class Tuesday, February 24:**

- Remember you should be prepared to present at least one proof in class and should work through all the exercises in the reading. If you cannot solve a proof, move on to the next Theorem and try again later. You can, of course, always ask me questions!
- Come to class ready to claim which theorems you are ready to present.
- Prepare a proof by induction of Theorem 2.1. Mark will be presenting it to us at the beginning of class on Tuesday.
- Read pages 29-33 of Chapter 2. Prove (or revise your proofs to!) Theorems 2.3 and 2.7. Then jump ahead and prove Theorem 2.27, before you prove Lemma 2.8 (Hint: Perhaps use Theorem 2.27!). Then try proving Theorems 2.9 and 2.18. From now on, assume that any Questions in the reading should be addressed the same way exercises and examples always are. That is, do ALL of them.

**Reading Assignment for class Thursday, February 26:**

- Remember you should be prepared to present at least one proof in class and should work through all the exercises in the reading. If you cannot solve a proof, move on to the next Theorem and try again later. There are some more straight forward proofs in this set, so everyone should be able to presnt at least two! You can, of course, always ask me questions!
- Come to class ready to claim which theorems you are ready to present.
- Now that you have a huge table of primes, experiment with evaluating $\pi(n)$ for larger numbers. Do you think you need to change your original conjecture or do these experiments confirm your previous ideas?
- Use proof by contraposition to prepare a proof of the second half of Theorem 2.3. Chris will do this for us at the beginning of class, but everyone should be prepared to do it!
- Review Euclid's Lemma and see why it can help you do a short proof of Theorem 2.27.
- Reread pages 31-33 of Chapter 2. Prove or revise your proofs to Theorems 2.9, 2.18, 2.19 and 2.20; and Lemma 2.8.
- Remember that a reading assignment includes working through all exercises, examples and QUESTIONS in the reading as well proving/solving the specified Theorems and Lemmas so that you are ready to ask questions about, discuss and present the material.
- Part of class will be spent in review for the exam on Sunday. Come to class prepared with questions.

**Quiz 2 will be on Thursday, February 19th. It will cover material beginning on page 16 through what we
cover in Tuesday's class.**

**Collected Homework (Due Monday, February 16 at 4:00pm):**

- Complete the problems on this handout.
- You may also resubmit the Notebook problems that were due February 2 or February 9. Note that you may turn in your resubmissions on any day of the week up to one month after they were originally due. Remember that you must turn in your previous draft(s) with any resubmission. Please do not write on your previous submissions after they have been returned.

**Reading Assignment for class Tuesday, February 17:**

- Remember you should be prepared to present at least one proof in class and should work through all the exercises in the reading. If you cannot solve a proof, move on to the next Theorem and try again later. You can, of course, always ask me questions!
- Reread pages 19-25 of Chapter 1. Continue working on Exercise 1.50 and be ready to present/discuss it. Prove Theorems 1.53, 1.57 and 1.58. Also address and be ready to present/discuss Questions 1.49 and 1.52. Remember that a reading assignment includes working through all exercises and examples in the reading as well proving/solving the specified Theorems and Questions so that you are ready to ask questions about, discuss and present the material.
- Start reading Chapter 2 [Primes!] (pages 27-29). Prove Theorems 2.1 and 2.3.

**Reading Assignment for class Thursday, February 19:**

- Remember you should be prepared to present at least one proof in class and should work through all the exercises in the reading. If you cannot solve a proof, move on to the next Theorem and try again later. There are some more straight forward proofs in this set, so everyone should be able to presnt at least two! You can, of course, always ask me questions!
- Come to class ready to claim which theorems you are ready to present.
- If you haven't done Blank Paper Exercise 1.59 yet, you should!
- Read/reread pages 27-32 of Chapter 2. Prove Theorems 2.1, 2.3, 2.7 and 2.9; and Lemma 2.8. Work on all of them, but really try to get a nice, complete proof of at least one of them.

**Due to needing to visit another professor's class this week, my office hours Monday and Wednesday of this week are altered. They are:
Monday, February 9th, 3:15-5:00PM and Wednesday, February 11th, 3:15-4:15PM. If you
cannot make these times and need to see me, please email me and make an appointment.**

**Collected Homework (Due Monday, February 9 at 4:00pm):**

- Complete the problems on this handout.
- Notebook Problem: Prove the following theorem, often called the Archimedean Property: If a and b are natural numbers, then there exists a natural number n such that na is greater than or equal to b. (Hint: You may want to use the Well Ordering Axiom.) (Remember: (1) Turn these in on a separate piece of paper from your other homework, (2) notebook problems must be done on your own, and (3) you should give extra attention to form for these proofs.)
- You may also resubmit the Notebook problem that was due Monday, February 2. Note that you may turn in your resubmissions on any day of the week up to one month after they were originally due. Remember that you must turn in your first draft with any resubmission, and that these are like take-home exams.

**Reading Assignment for class Tuesday, February 10:**

- Remember you should be prepared to present at least one proof in class and should work through all the exercises in the reading. If you cannot solve a proof, move on to the next Theorem and try again later. You can, of course, always ask me questions!
- Read pages 19-25 of Chapter 1. Prove Theorems 1.38, 1.40, 1.41, 1.43, 1.45 and 1.48. Also address Questions 1.44, 1.46, 1.47 and 1.49. Remember that a reading assignment includes working through all exercises and examples in the reading as well proving/solving the specified Theorems and Questions so that you are ready to ask questions about, discuss and present the material.

**Reading Assignment for class Thursday, February 12:**

- Remember you should be prepared to present at least one proof in class and should work through all the exercises in the reading. If you cannot solve a proof, move on to the next Theorem and try again later. There are some more straight forward proofs in this set, so everyone should be able to presnt at least two! You can, of course, always ask me questions!
- Come to class ready to claim which theorems you are ready to present.
- Reread pages 19-25 of Chapter 1. Prove (or check your previous proofs for) Theorems 1.41, 1.43, 1.45, 1.48, 1.53 and 1.57.
Also address and be ready to present/discuss Questions 1.44, 1.46, 1.47, 1.49 and 1.52.
**Be sure you really work on these Questions before you continue to the next part of the text!**

**Collected Homework (Due Monday, February 2 at 4:00pm):**

- Complete the problems on this handout.
- Notebook Problem: Prove that for any integer x, one of the integers x, x+2 and x+4 is divisible by 3. [Remember: (1) Turn these in on a separate piece of paper from your other homework, (2) notebook problems must be done on your own with no outside resources other than me, and (3) you should give extra attention to form for these proofs.]

**Reading Assignment for class Tuesday, February 3:**

- Remember our goal is to work through these without outside resources. After we have tried to develop material on our own, then we can discuss utilizing outside resources. For example, you are to derive the Euclidean Algorithm on your own in the assignment below, not look for someone else's derivation. You have been showing great work in class!
- Read pages 16-20 of Chapter 1. Prove Theorems 1.27, 1.28, 1.33, 1.38 and 1.40. Note that you may assume that any previous theorem has been proved when you attempt the next proof. Also address Questions 1.29 and 1.30. Note that Exercises 1.35-1.37 are very important! They are asking you to develop the idea behind the Euclidean Algorithm. Try to write as explicit an algorithm as you can. Remember that a reading assignment includes working through all exercises and examples in the reading as well proving/solving the specified Theorems and Questions so that you are ready to ask questions about, discuss and present the material.

**Quiz 1 will be on Thursday, February 5th. It will cover material beginning on page 7 through what we
cover in Tuesday's class.**

**Reading Assignment for class Thursday, February 5:**

- Come to class ready to claim which theorems you are ready to present.
- Read pages 18-21 of Chapter 1. Prove Theorems 1.33, 1.38, and 1.40-1.43. Keep working on getting your proofs refined to present in class. You are doing great! Please feel free to come and discuss them with me. Note that you may assume that any previous theorem has been proved when you attempt the next proof. Also address Questions 1.44 and 1.46. Remember that a reading assignment includes working through all exercises and examples in the reading as well proving/solving the specified Theorems and Questions so that you are ready to ask questions about, discuss and present the material.

**Remember to keep your appointments!!!**

**Collected Homework (Due Monday, January 26 at 4:00pm):**

- Write a proof of Theorem 1.3 and answer Question 1.4 on pages 9-10 in our text. Note that to justify your answers to Question 1.4, you need to write proofs (either counterexamples or direct proofs)!
- Complete Problem 3 from the first groupwork sheet: As we have discussed, number theory has a lot to do with playing with numbers and looking for patterns. Find the sum of the first two odd natural numbers. Find the sum of the first three odd natural numbers. Continue for several more sums and compare your answers. Can you find a pattern? Propose a conjecture and then see if you can prove it. What is the best method of proof for this theorem?
- There will be no Notebook Problem with this assignment. Remember that although you may discuss this assignment with others, your write up should be your own.

**Reading Assignment for class Tuesday, January 27:**

- We will start Tuesday's class with Thomas's presentation of the proof of Theorem 1.14. Everyone should be sure to have his/her own proof!
- Read pages 12-17 of Chapter 1. Prove Theorems 1.18, 1.21, 1.22 and 1.23 (note that 1.22 and 1.23 are the two parts of the theorem listed at the top of page 14). Also address Questions 1.20, 1.29 and 1.30. If you have time, also tackle Theorems 1.26 and 1.27, which are the two parts of the Division Algorithm Theorem. Remember that a reading assignment includes working through all exercises and examples in the reading as well proving/solving the specified Theorems and Questions so that you are ready to ask questions about, discuss and present the material.

**Note that due to introductory appointments with my MATH 135 students and you all, some of my open office hours are shortened this week. The
revised hours for Wednesday are: 2:40-4:00. If you cannot make these times and need to see me, please make an appointment.**

**Reading Assignment for class Thursday, January 29:**

- Note some of this is a repeat of the previous assignment! :-) Read pages 14-18 of Chapter 1. Prove Theorems 1.22 and 1.23 (note that 1.22 and 1.23 are the two parts of the theorem listed at the top of page 14), and also Theorems 1.26 and 1.27 (really two parts of the Division Algorithm Theorem), and Theorem 1.28. Note for 1.26 the hint suggests that you define q and then choose r from there, but it is possible to do it the other way around. Also address Questions 1.29 and 1.30. What are the authors getting at with these questions? Remember that a reading assignment includes working through all exercises and examples in the reading as well proving/solving the specified Theorems and Questions so that you are ready to ask questions about, discuss and present the material.

**Welcome to Number Theory!!!**

**Collected Homework (Due Thursday, January 22nd at 1:30pm):**

- Write an autobiographical essay or poem as assigned on the syllabus. Sign up for an appointment to meet with me when you drop off your essay. The sign-up sheet should be on my bulletin board.

**Reading Assignment for class Thursday, January 22:**

- Read the syllabus at least once!
- Check that you have no conflict with the dates and times of the midterm exams. Speak with me ASAP if you do have conflicts.
- Familarize yourself with this website. Note that there is a link at the top of the page to our syllabus, should you lose the orange one I handed out in class. The syllabus has a lot of vital information on it and you will likely want to refer back to it regularly. Also at the top of the page is a link to my grade scale. This will let you know what percentage you need to earn in order to obtain specific grades. In addition, there is a website I developed, Proof Writing and Presentation Tips, for my First Steps into Advanced Mathematics classes. It would be valuable for you to use this as a reference when you are preparing your homework and presentations for class. Lastly, there are some fun websites about numbers!
- Make sure you have worked through Exercise 2 from the class handout before you start reading the text.
- Read the introduction chapter (pages 1-3), and pages 7-13 of Chapter 1. Address Question 1.5 and prove Theorems 1.6, 1.9-1.14 and 1.18. If you have time, feel free to keep going!!! Remember that a reading assignment includes working through all exercises and examples in the reading as well proving/solving the specified Theorems and Questions so that you are ready to ask questions about, discuss and present the material.