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Math 311: Number Theory

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Course Materials:

Syllabus
[uploaded 07.14.15]
Project Topics
[uploaded 07.14.15]
Textbook Project
[uploaded 08.11.15]
Textbook Template
[uploaded 07.16.15]
Portfolio Template
[uploaded 08.06.15]
Presentation Rubric
[uploaded 11.16.15]

Course Log:

Week 15
[12.07.15]

Thanks to all of you for a great semester. Here is a PDF of our course textbook, on which you all did a fantastic job.

Week 13
[11.16.15]

Today we proved Theorems 6.12 (h/t Taryn and James) and 6.13. Which means we finally know that primitive roots exist! Wednesday will be our last normal class of the semester, we’ll finish up problem set 6.2, and I’ll try to tie up any other loose ends of the semester.

Now it’s time to start thinking about the Textbook project. Today I assigned groups for the project, if you weren’t in class today, be sure to get in touch with your group members. Your individual chapter is due on Tuesday Dec. 1st, and you can find the template in the course materials above.

[11.18.15]

We made it to our last day of number theory content. You all championed this “befuddling” subject, the so-called “queen of mathematics.” This week will be your last portfolio submission. Please include proofs of theorems 6.14 and 6.15, and in your narrative summary, please address how you would answer the questions: What is number theory?

Beginning the Monday after Thanksgiving we will start our presentations, check the rubric above for preparing your presentation. It should be 20 minutes long, and a good format to follow is 5 minutes of introduction, 10 minutes of content, 5 minutes further directions, fun facts, etc. If you are making slides, you should shoot for no more than 15 slides. We may need to stay a few minutes over each day, if that’s a problem, please let me know. The order of presentations is as follows:

  • Monday Nov. 30th: Autumn, Michelle, Marina, Josh
  • Wednesday Dec. 2th: Brad, Ashleigh, Allie, Taryn
  • Monday Dec. 7th: James, Maggie, Alex

Week 12
[11.11.15]

Greetings from Rhode Island! It sounds like you have been making some steady progress in my absence; thank you so much for your hard work. I heard that proofs were presented by Allie, James, Josh, and Alex (x 2) on 11.09.15 and by Josh and Alex on 11.11.15.

For Friday this week, you will have a homework assignment due, again coming from homework’s 4, 5 or 6. You will also have a portfolio due, it should include proofs of the Lemma, 6.10 and 6.11. Please be very careful with your write-up, since I haven’t had the chance to see these proofs yet.

Week 11
[10.02.15]

Today we slogged through a few proofs off of problem set 6.1. Josh started out by giving us a proof of 6.3 using induction, Alex was able to mostly salvage it, but by the end of class it seemed there was still a hole. We did have a successful proof of 6.4 courtesy of Alex.

For the next class Michelle and Allie will come prepared to present a proof of 6.3 and 6.5. If anybody has a proof of 6.6 you are welcome to present it, but I am also happy to do this one for you guys. After that we can start talking about the meaning of these proofs in the context of primitive roots.

[10.04.15]

Great work today! We nailed down our proofs of 6.3 and 6.5 and are getting closer to establishing the existence of primitive roots. Next week I’ll be in Rhode Island at the Institute for Computational and Experimental Research in Mathematics at Brown University working on the LMFDB Database of number theoretic objects (it’s kind of like Facebook for numbers). Dr. Kern will be taking my place on 11.9 and 11.11, and I hope that you all impressing him with your dazzling proof writing skills. For Monday, please come prepared to present proofs of 6.7-6.11 and the Lemma:

For p prime and an integer k, Φ(pk)=pk-1(p-1).

I will continue to update the website while I am away.

For your portfolio this week, please include a definition of primitive root modulo n and proofs of theorems 6.3, 6.4 and 6.5. Also, if you haven’t done so already, please let me know your chosen project topic.

Week 10
[10.26.15]

Scavenger hunt day was so much fun! Congrats to the winning team, Michelle, Taryn, and Lisa for bringing my beautiful marker mug out of the clutches of Dr. Kern and back into the safety of my office. And bravo to all the other teams that came so close; you guys all did a commendable job.

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On Wednesday we will resume with our regularly scheduled programming of problems sets and presentations. Though there is no portfolio this week, there will be a homework assignment due. This week your problems must come from Homework 4-6.

[10.28.15]

Today we redefined primitive roots modulo n, are started exploring what it means to be a primitive root. In particular, Maggie and Autumn proved theorems 6.1 and 6.2, and we computed some examples (and non-examples) of primitive roots when n is 3,4,5, or 8. For Monday, please be prepared to present 6.3 and 6.4, and recall that you have a homework due this Friday which must come from Homework 4-6.

Week 9
[10.19.15]

Today we finished up problem set 4.2 with proofs of 4.18 and 4.19 from Taryn and James. Now it’s time to get started on the cryptography part of the semester. This is the imd when we start to see some of the fruits of our labor in action, from basic modular arithmetic in the Caesar cipher, to the more sophistical Euler’s theorem based RSA encryption. Autumn and Marina presented proofs of 5.1 and 5.2 today to give us a basis for the RSA encryption algorithm.

For next time please come prepared with your own public and private RSA key, that is, your own choice of (n,E) and (n,D) from problem set 5.1. On Wednesday you’ll be encrypting messages to send to each other!

[10.21.15]

Today we tried out some RSA encryption with great success. We also learned about two new methods of encryption: the polyalphabetic caesar cipher and Diffie Helman key exchange. Since we’ve started dealing with moduli of really big numbers, it will be helpful to start using some kind of computer system to compute them. The best thing I’ve found is the Wolfram Alpha App (which is free for iPhone, but maybe costs a few dollars for android). Over the weekend, look around for some kind of app that can help you compute big moduli, and on Monday come prepared to do computations on a smartphone, tablet, or laptop of your choice. Also, bring your student ID.

There will be no portfolio due this week. Keep in mind that you need to choose your project topic before Thanksgiving. If you need inspiration on that front, come and see me!

Week 8
[10.12.15]

Today we finished up the proofs from problem set 4.1, concluding with a proof from Maggi of Fermat’s Little Theorem. We discussed using the F.l.T as a test for primality, but noted that this is not actually a very good test since there are some strange numbers that look prime based on the F.l.T., but actually aren’t. If you’re interested in things like this, you might consider psuedoprimes or Carmichael numbers for your end-of-semester research project.

For Wednesday, please come prepared to prove conjectures 4.13-4.15, and recall that you will have a homework assignment due this Friday. This means two homework problems of your choice. But since Exam 1 is behind is, now your problems must come from Homeworks 4-6.

[10.14.15]

Today we proved theorems 4.13-4.17, including a corollary to 4.16 which we will call Corollary 4.16.1: Inverses are unique modulo n. Thanks to everyone for coming prepared today, and thanks to Michelle, Josh, Marina, Brad, Ashleigh, and James for sharing your proofs with us. We finished up class with some number theory gossip including several examples of Stigler’s Law of Eponymy in elementary number theory. On Monday we will finish up problem set 4.2, please come prepared to present 4.18 and 4.19. Once that is done we will begin out tour into the world of cryptography.

For Friday this week you have a bi-weekly homework due. You will also be submitting a portfolio which will contain definitions for Zn, Znx, Euler’s phi function, and inverse modulo n, as well as proofs of theorems 4.11, 4.17 and Corollary 4.16.1.

Week 7
[10.07.15]

Exam 1 is behind us; onward and upward! Today we carried on deeper into our inquiry of orders modulo n. We spent the bulk of class time working in small groups, and at the end of class Ashleigh gave a proof of 4.6. We ran out of time today, but on Monday Allie, James and Alex will present the proofs below at the beginning of class. If you’d like to try to pick up some extra presentation points, you can feel free to get started on the proofs on Problem Set 4.2. But you will need a few definitions to get started: Z_n is just a fancy notation for the canonical residue system, that is, the set {0,…,n-1}. And when we add the upper x, we mean the set of all elements in {0,…,n-1} which are relatively prime to n.

For your portfolio this week, please include a proof of 4.6 and 4.10. Although, technically we will not have proved 4.7-4.9 as a class, you can feel free to cite them in your proof of Fermat’s Little Theorem. A note on portfolios, I keep seeing two mistakes being repeated:

  • Any time you talk about a set of numbers it should be enclosed in curly brackets not parenthesis (i.e. {0,…,n-1} rather than (0,…,n-1), which means something different in mathematics).
  • Always enclose ALL mathematical symbols on dollar signs, whether numbers, variables, mod. If it’s math, then put it in math mode.

Week 6
[09.28.15]

Today we finished up our tour of modularity and diophantine equations and learned a bit about Pierre de Fermat, Andrew Wilesm and Fermat’s Last Theorem. Michelle gave us a great proof of 3.13, which gave people some trouble on last week’s portfolio, and James proved Lemma 2, setting us up to prove the Chinese Remainder Theorem.

We started in on Problem set 4.1 and learned the definition of order n. For next time, please be prepared to present proofs for conjectures 4.1, 4.2 and 4.3. Also, recall that Exam 1 is on Monday next week and will come directly off of Homework 1,2 and 3.

[09.30.15]

Today was all about proving things in small groups. Taryn and Brad started us off with proofs of 4.1 and 4.2, and Allie gave us a partial proof of 4.3 which was eventually salvaged by Alex. Thanks to all of you who came prepared with proofs today. Based on the group work discussions today Autumn and Lisa were able to present proofs of 4.4 and 4.5. As an upshot to our discussions today, we learned the importance of proving that the rings we define actually exists. Because what’s the point of defining an order if no element even has one?


This Friday I will be collecting your bi-weekly homework assignment, remember you only need to do two problems off of any homework of your choice and the write-ups can be far more casual than your portfolio. Your portfolio this Friday should contain a definition for order of a modulo n and proofs of theorems 4.3, 4.5. And don’t forget (as though you possibly could!) there is an exam on Monday!

Week 5
[09.21.15]

Today we learned that we can solve really hard divisibility problems in just a few easy steps using modular arithmetic. For example, we showed that 41 divides 2^20-1 without actually computing that gigantic number. We also proved Theorems 3.7, 2.8 and 3.9, with thanks to Josh, Ashleigh and Autumn, and James.

Theorem 3.7 gave us an interesting result. Namely that any time we have integers a and n, we can always find an integer between 0 and n to which a is congruent modulo n. This is a very important idea in congruence arithmetic, one which we will explore further next time. Please come prepared to present proofs of 3.10 and 3.11, and remember that you have an exam coming up on 10.05.15. Also, for those interested, I’m giving a seminar talk on Wednesday at 2:00 in room 446, it’s called “17 facts about science writing that will totally blow your mind.” It will be fun. And mind-blowing. Obviously.

[09.23.15]

Today talked about different ways to solve a congruence for x, with the help of theorems 3.10, 3.11, and 3.12, proved by Ashleigh, Marina and Alli.

This week you will be submitting a portfolio with a narrative of the week’s mathematical content, definitions for residue system and canonical residue system, as well as proofs of theorems 3.7 and 3.13. Remember that the statement for 3.13 has been slightly appended, it now reads “…mod n has no solutions, or has precisely d mutually…” Also, for monday, be prepared to prove Lemma 2 from class today.

Week 4
[09.14.15]

Today we gossiped about primes, from the prime number theorem, to Zhang and Maynard’s work on the Twin Prime Conjecture, to Frank Nelson Cole’s epic counterexample. You can find the slides right here. Remember, if you are interested in exploring any of this more for your research project, just let me know. A few different things came up today that would make great project topics.


Thanks to James and Marina for coming prepared with proofs for Theorems 2.5 and 2.6. By the end of class we even had complete proofs for 2.7 and 2.8 from Brad and Taryn, and Alex got us started on 2.9. For next time, be prepared to present the rest of the problems on the sheet when we get to class. And remember, you will have a homework assignment due this Friday.

[09.16.15]

Today we finished off Problem Set 2.2, ending our tour de primes. Theorems 2.9-2.12 were presented by Ashleigh, Michelle, Brad and Josh. Moving on to Problem Set 3.1 we defined congruence modulo m and tackled theorems 3.1-3.4 and 3.6 faster than a Steelers linebacker brings down an…anybody who is not a Steeler. Thanks to Marina, Maggi, Jess, Alex, and James for that.


On Friday you have two homework problems due, these can be hand written or typed, and either emailed to me or slipped under my door anytime before Friday evening. In addition you will have your weekly portfolio due. This week you will have your usual narrative summary (remember this should be an explanation of the math we did and how the theorems are connected), definitions for arithmetic progression, factorial, and congruence modulo m, and proofs of Theorems 2.9 and 3.5.

For Monday, please come prepared to present Theorems 3.8, 3.9, and 3.10 from Problem Set 3.2. Also, a friendly reminder that your first exam will be on Monday October 5th. Thanks again for all of your hard work this week!

Week 3
[09.09.15]

Today was PRIME TIME!!! We defined the primes and discussed different ways for deciding which numbers are prime, we even worked through the ancient sieve of Eratosthenes. Allie, Josh and James gave us some very nice proofs of theorems 2.1-2.3.


For Friday, you will be submitting your weekly portfolio and it should contain definitions for prime, composite, and prime factorization, as well as proofs of conjectures 2.3 and 2.4. For an example of what I’m looking for in the narrative, check out this sample narrative. To start class on Monday I’ll be asking for volunteers to prove theorems 2.5 and 2.6 — let’s have the volunteer be YOU. Stop by if you need a gentle nudge in the right direction.

Week 2
[08.31.15]

Today we established the Euclidean Algorithm for finding the greatest common divisor of two integers, a and b. Then we used our “scrap material” from the Euclidean algorithm to find solutions to the equation ax+by=1. Using the principles we developed, Alex was able to prove Theorem 1.6. — way to go, Alex!

For next time, you should be prepared to present proofs for conjectures 1.7-1.11 in the beginning of class. And remember, this Friday you will be turning in a portfolio with a selection of proofs from Problem Set 1.2, as well as two homework problems of your choice. You may want to use Overleaf for your portfolio, and you can feel free to use old fashioned pencil and paper for the homework problems.

[09.02.15]

Whew, what a day! Things started off with a splash today when Taryn, Michelle, and James gave us proofs for Theorems 1.7, 1.8, and 1.10. Things got a bit hairy with Theorem 1.9, but eventually Brad was able to salvage his partial proof with the help of Lemma 1.9.1, proved for us by Jessica and Michelle. Alex also gave us an alternative proof of 1.9 —pick your favorite! Finally, in a combined effort, Autumn and Allie gave us a proof of 1.11. Today was tough, by everyone did a great job.


For our next meeting, which isn’t until Wednesday 09.09.15, you should be prepared to present theorems 2.1, 2.2, and 2.3 off of Problem Set #2.1. For Friday this week I’ll be collecting two homework problems in addition to your weekly portfolio. This week’s portfolio will contain a narrative summary of the mathematical content of the week, along with proofs of theorems 1.9, 1.12, and 1.13.

Week 1
[08.24.15]

Today we discussed the merits and veracity of a variety of definitions for axiom, theorem, and proof, and established a working list of axioms for the integers. Next time we’ll be working on problem set 1.1 in class and presenting proofs.

Remember for Wednesday you should try to set up LaTeX — this website gives a good step-by-step tutorial for installing and setting it up, and this youtube channel has some helpful videos. If you have trouble, bring your laptop to class on Wednesday.

[08.26.15]

Today we learned about the division algorithm and saw a proof that established the existence and uniqueness of the quotient and remainder terms. Next we were treated to some flawless proofs of theorems 1.1, 1.2, 1.3, and 1.4 by Ashleigh, Maggi, James, and Bradley — thanks you guys!

On Friday 08.28.15 you will be submitting your first portfolio, per the instructions in the Textbook Project Guidelines above. If you’re having trouble setting up TeX, come by my office with your laptop and I can help you troubleshoot. This week’s portfolio should include a short narrative summary of what your impressions are of the course so far, definitions for axiom, theorem, proof, integers, natural numbers, divides, greatest common divisor, and relatively prime, and proofs of theorems 1.4 and 1.5.

Week 0
[07.14.15]

Before we get started, I thought I should explain the unusual setup of this course. We’re going to be learning number theory through Inquiry Based Learning (IBL), sometimes called the Modified Moore Method. Why are we doing this? Well, researchers have a lot of positive things to say about IBL; things you learn in an IBL course stick in your brain for longer, and the meta-skills you acquire in an IBL course will actually help you do better in future math classes — even if they aren’t number theory!

The day-to-day activities of this course might feel strange and unfamiliar at first, but I’ll do my best to keep us all on the right track. I’ll be documenting our daily activities and successes on the Course Log, and I’ll use this as a place to keep everyone in the loop on due dates. I’m looking forward to a great semester!

By the end of Week 1 you will need to have LaTeX set up on your computer. This website gives a good step-by-step tutorial for installing and setting it up, and this youtube channel has some helpful videos.

Homework Assignments:

Homework 1
[uploaded 07.26.15]
Homework 2
[uploaded 07.26.15]
Homework 3
[uploaded 07.26.15]
Homework 4
[uploaded 07.14.15]
Homework 5
[uploaded 07.14.15]
Homework 6
[uploaded 07.14.15]