Math 453 C3M Home Page

This will be the home page for Math 453 D13 "Elementary Theory of Numbers". This class meets for the Fall 2019 semester at MWF 11:00-11:50 in 343 Altgeld. Office hours to be announced.
The mathematical links page will be here. Suggest a new one!

Many students are concerned about the political issues of the day. If you want to contact your representatives, here are their numbers:
US Sen. Tammy Duckworth (D-IL) (202)-224-2859
US Sen. Richard J. Durbin (D-IL) (202)-224-2152
US Rep. Rodney Davis (R-IL 13) (202)-225-2371
IL Sen. Scott Bennett (D-IL 52) (217)-335-5252
IL Rep. Carol Ammons (D-IL 103) (217)-531-1660

F 1/3 -- This webpage will disappear before the end of January 2020. Write me if you need anything that was on it.

M 12/23 -- Report. You were an excellent class! On the Final, the distribution was: 200 or more (4), 190s (5), 180s (6), 170s (4), 160s (4), 150s (3), 140s (2), less than 140 (2). The course grades were A+ (4), A (10), A- (4), B+ (3), B (2), B- (1), C+ (2), C (3), D(1). The most popular questions to skip were #15, #10, #8, and #12 (in that order). Most common error was using the Pythagorean parameterization where it didn't apply. Have a great break! If you want to learn your specific information, please email me with your booklet number as PIN.

FINAL information: In class Th 12/19 1:30-4:30 in the usual room. The exam will consist of 4 20 point problems and your choice of 8 pf 10 15 point problems. There will be roughly equal weight placed on each third of the course. I will leave a short pile of index cards on my door, in case you didn't pick one up. You can use your own card, as long as it's 3" x 5" and as long as everything on the card was written by you.

W 12/11 -- An extended set of solutions for the 12/9 worksheet, with additional discussion of Diophantine equations, factoring and applications of congruences. The ICES forms were distributed. A beer can was signed.

M 12/9 -- Test 3 and HW 10 returned. Some discussion on Diophantine equations, to be continued W 12/11 and a worksheet. (Solutions, and more will be distributed 12/11.) Also a very brief introduction to RSA cryptography, and how it fits in with this course.

F 12/6 -- Test 3. Not online.

W 12/4 -- Here are major topics for the third test, on Friday. I did some examples, including a "lifting" of the primitive root 2 for 5 up to 25. (Works for 2,7,12,17, not 7.) Also an inelegant presentation that there are no non-trivial solutions to x^2 + y^2 = 7 z^2, and the proof works when 7 is replace by any prime p = 3 mod 4.

M 12/2 -- Distributed the Tenth Homework Solutions and a handout on the geometric method for solving some quadratic Diophantine equations. 3 x 5 cards were distributed for Test 3, on 12/6. We talked about writing integers as a sum of two squares, which is section 6.5 in the book.

F 11/22 -- A freeflowing discussion of a lot of different topics, most related to mathematics, research and teaching, rather than 453, although I did distribute the solutions to the worksheet of 11/20.

W 11/20 -- Homework 9 returned. More on Pythagorean triples, if I remember right, plus the geometric method and there was a worksheet. (Solutions are in 11/22,)

Homework 10

  • Homework 10 is now due on Monday, Dec. 2, the day after Thanksgiving break, instead of Friday, Nov. 22, so we can have more time to talk about the subject.

    M 11/18 -- Homework 8 returned. We moved onto Diophantine equations. Important announcement on the due date of HW 10, see above.

    F 11/15 -- Passed out solutions to 11/13 handout (see 11/13) and the Ninth Homework Solutions and the Tenth Homework, due F 11/22. This is the final graded homework of the semester. The Third Exam will be on F 12/6.

    W 11/13 -- All primitive roots, all the time. There was a handout; here are solutions.

    M 11/11 -- Test Two returned and discussed. We are fully into chapter 5 now. Ask more questions!!

    F 11/8 -- More on quadratic reciprocity, with a transition to Chapter 5 on Primitive Roots. The Eighth Homework Solutions were distributed. The Ninth Homework was distributed, due F 11/15

    Here is the Ninth Homework. I will fully update, with HW 8 solutions, tomorrow. No office hour Monday because of the Senate meeting; office hour on Tuesday at 5 is cancelled due to the weather.

    W 11/6 -- Examples of Quadratic Reciprocity, plus a handout of two problems, each done two ways. There was a mistake on p.2 of the handout, which is corrected here. Also, a rare Wednesday Worksheet, and its Solution, distributed early for your homework solving convenience.

    M 11/4 -- Examples of Quadratic Reciprocity in action. There will be a worksheet on Wednesday on this. The Eighth Homework was distributed, due F 11/8.

    F 11/1 -- Test 2. Not online.

    W 10/30 -- Here are major topics for the second test, on Friday. Lots of review and clarification. We finished the proof of quadratic reciprocity. I'll post a copy of the handout later.

    M 10/28 -- More review of multiplicative problems and progress on quadratic reciprocity. Here is a solved version on the last worksheet.

    F 10/25 -- The Seventh Homework Solutions were distributed.There was also a Worksheet distributed at the end of class. Most of the class was spent reviewing HW 7, with other approaches and lots of questions. Review in advance of test 2 will take priority next week. Bring questions, or send them by email.

    W 10/23 -- More on quadratic reciprocity, reviewing proofs and examples. This is tough stuff!


  • A line was cut off of problem 4&5 d. Noted by an alert student. Thanks! After the formula, please add "if n = p_1...p_r, and (Q*mu)(n) = 0 if n has a repeated prime factor. "

    M 10/21 -- The second hour exam will be on F 11/1. Same pattern as the first, but no scratch paper except the back of the sheets. A few review questions on multiplicative functions, plus distribution of the filled-out worksheet. The main topic was a full-force introduction to Quadratic Reciprocity.

    F 10/18 -- Mostly review. The Sixth Homework Solutions were distributed. The Seventh Homework was distributed, due F 10/25. There was also a Worksheet distributed at the end of class; here are the Solutions, which will be distributed in class 10/21.

    W 10/16 -- Continuation, with more lecture notes. The instructor intuited that he was enjoying this material more than the class.

    M 10/14 -- A "higher level" approach to arithmetic functions, based on the following lecture notes. These will be extended on Wednesday.

    HW 6

  • An alert student points out that the answer in 5b (second line) should have been requested mod p(2p+1), and not mod 253!

    F 10/11 -- Two links. The first is to a video on patterns of putting points in polar coordinates that naturally brings up some of the topics of the course so far. The second is to the land acknowledgement, as presented by the Chancellor at the start of every Senate year, and inspired here by the upcoming "Columbus Day".

    F 10/11 -- The Fifth Homework Solutions were distributed. The Sixth Homework was distributed, due F 10/18. There was a Worksheet distributed at the end of class. There will be notes on Monday as we go to a more formal approach to arithmetic functions.

    W 10/9 -- First exam returned and gone over. More on arithmetic functions, including a more comprehensible proof perhaps on even perfect functions.

    M 10/7 -- Back to the course. I hope to have the first test graded and back on W. Finished talking about the Euler phi function and introduced sigma_k(n) = \sum_{d | n} d^k. Special cases in the book are sigma_0(n) = d(n) and sigma_1(n) = sigma(n). You didn't believe my proof on even perfect numbers. I'll try again on Wednesday!

    S 10/5 -- Here is the Fifth Homework, which will be distributed in class on Monday. Here is a link to the last spring's course description of Math 496. I expect to teach it in the spring, but nothing is definite yet.

    F 10/4 -- Test 1. Not online.

    W 10/2 -- Key terms and properties for the first test, on Friday, were discussed and various problems were worked through. No new material was presented.

    M 9/30 -- Solutions to Friday's worksheet were distributed. A few questions on the Chinese Remainder Theorem were discussed, and more fun facts about the Euler phi function and its unexpected connection with pi. On W 10/2, I will give a list of key terms and properties to help orient your studying. You are allowed to bring one (1) 3 x 5 card (distributed on M; I'll bring extras on W), and you can write on both sides if you like, but no books, no notes and no electronic devices.

    F 9/27 -- The first test is a week from today, so HW 5 will be delayed, and due F 10/11. The Fourth Homework Solutions were distributed, and problems from HW 3 and HW4 were discussed. More on the Euler phi function, in great detail. There was a Worksheet distributed at the end of class, solutions to be discussed on M 9/30.

    W 9/25 -- We spent more time talking about multiplicative and completely multiplicative functions, and began to focus on the Euler phi function.

    Homework 4 comments and corrections

  • #5 I fixed this in class, but the minus was meant to be a dash. What I wrote on the board was "Determine the last two decimal digits of 347^453; that is, find 347^453 mod 100. Note that 100 = 2^2*5^2..."

    M 9/23 -- Solutions to Friday's worksheet were distributed. We went through the definition of the Euler phi function in great detail. This concluded the discussion of material for the first test. We began to talk about multiplicative and completely multiplicative arithmetic functions.

    F 9/20 -- Review of a lot of things. We are in the middle of Euler's Theorem, which I will talk about in greater detail on M 9/23. The Third Homework Solutions were distributed. The Fourth Homework was distributed, due F 9/27. There was a Worksheet distributed at the end of class, but we didn't have time to do it. I will distribute a solved version of it in class on M 9/23.

    Homework 3 comments and corrections

  • Yes, there's a typo in #7(ii). That should be r, s > 2. The problem is false in case, e.g. r=2, s = 3. My apologies.

    W 9/18 -- The first hour test is scheduled for Friday, October 4. Wilson's Theorem and Fermat's Little Theorem were the main topics; there was a review of what we've been doing with solving ax = b mod m and the Chinese Remainder Theorem, and a bio sketch of former UIUC prof (1915-1947) Robert Carmichael (1879-1967), who discovered Carmichael numbers.

    M 9/16 -- The filled-in worksheet was distributed. This contained a solution to another CRT problem. In class, I went through the proof of the CRT in excruciating detail, solved Sun Zi's problem in two ways, and got close to proving Wilson's Theorem, and sections 2.4 and 2.5. Don't be shy about emailing me questions.

    F 9/13 -- The Chinese Remainder Theorem. There was a related Worksheet which occupied the end of the hour. The Second Homework Solutions were distributed. (#4 is corrected on this!) The Third Homework was distributed, due F 9/20. On Monday, continue through the Chinese Remainder Theorem and its implications.

    W 9/11 -- Lots of examples of congruences and the solution of congruential equations, with data from the class. We are about to start the Chinese Remainder Theorem. Here is a biography of Sun Zi.

    M 9/9 -- Phone and email list distributed (no links), also a first handout on v_p(n) and its applications. We started in on congruences from sections 2.1 and 2.2 and are in the middle of the proof of Theorem 2.6, having already done so when gcd(a,m) = 1.

    F 9/6 -- Phone and email list circulated for the second and last time. GCD's and the graphical interpretation of divisibility of numbers of the form 2^a3^b. There was a related Worksheet which occupied the end of the hour. The First Homework Solutions were distributed. The Second Homework was distributed, due F 9/13. On Monday, we start Chapter 2 of the book.

    HW 1 Comments

  • If you want to see more examples of the Euclidean algorithm, go to the examples here. You can ignore the coding.

    W 9/4 -- We covered 1.5 up to p.28, plus problem 42 (p.21) with proof. I didn't get to the lcm, but I did define v_p(n), which is the exponent of the power p in the prime factorization of n. Prop. 1.17 and Thm. 1.19 are much "cleaner" that way. Friday will see the lcm, more on v_p(n) and a shortish worksheet at the end of the hour. Homework 1 will be due, and, I hope, Homework 1 solutions and Homework 2 questions will be distributed.

    F 8/30 -- Office hours will be Tues (5-5:30) in 443 AH and, most weeks, Mon (4-4:30) in 447 AH, except when the Senate is in session. The gcd and the Euclidean algorithm, with examples. I phrased Prop. 1.11 as aZ + bZ = gZ, where g = gcd(a,b). Next stop: the Fundamental Theorem of Arithmetic about unique factorization of positive integers into products of primes. The First Homework was distributed, due F 9/6.

    W 8/28 -- Talked about tentative test times. More explanation of f(x) = [x] (the floor function), and a lot of discussion about primes, both historical and otherwise. Most of it is in the book. The Prime Page is in the Fall 19 Links page cited above. Homework 1 will be out tomorrow and due 9/6. Also, office hours will be discussed.

    M 8/26 -- Introductions. Handouts were: Course Organization, How to Solve It guide, Class questionnaire. and information about the links page (see above.) We covered basically pp.1-10 of the text. Let me know if you have any questions. Notations not in the book: for a set A, x + A = {x+a:a in A}, for sets A, B, A + B = {a+b:a in A, b in B} (I forgot to give that one), and for n > 0, D(n) = {d : d | n}. We agreed to have homeworks due on Fridays. Let me know if I forgot anything. To answer an email question, the sums in A+B are taken over all elements independently, so if A = {1,2} and B = {10,20,30}, then A+B={11,12,21,22,31,32}.