Course Description -- Math 467 -- Fall 2016

Wednesday, 5:00 - 8:00, SEO 600

Instructor: Louis H. Kauffman

Office: 533 SEO

Phone: (312) 996-3066

E-mail: kauffman@uic.edu

Web page: http://www.math.uic.edu/~kauffman

Office Hours: 3PM to 4PM on MWF.

This is a course on elementary number theory. We will begin by recalling basic arithmetic, looking at it all over again and seeing how it works.

See ContFrac. This is a calculator on the web that converts numbers to their continued fractions. Try it on sqrt(2), sqrt(3), e, pi and other favorite numbers.

See PrimeFactor. This is a calculator on the web that factors numbers.

See Fractions and Decimals . This is a great page on fractions and decimals including a calculator for converting fractions to decimals and finding the perionds.

See Fraction to Decimal Converter . This is a sequel to the page above with a specific fraction to decimal converter.

See Roots of Two. This is a Wiki about the relationship of the twelfth root of two and the musical scale.

See Ulam Spiral 1. This is a Wiki about the structure of the primes and the Ulam Spiral.

See Ulam Spiral 2. This is a demo about the structure of the primes and the Ulam Spiral.

See Ulam Spiral 3. This is a YouTube Video about the structure of the primes, Ulam Spiral and the prime 41.

See Pebble Pythagoras. This is a proof that the square root of two is irrational via dot patterns or as the author of this book say, by "pebbles".

See Vi Hart. This is a link to math videos by Vi Hart.

See Time Line. This is a time line for the history of past and present mathematics. It is by no means complete, but you can use it to locate topics and to find when some ideas and results happened.

Recommended Book: "What is Mathematics?", Oxford Univ. Press, by Richard Courant and Herbert Robbins. This book is available at the UIC bookstore and also via Amazon.

Recommended Book: "The Heart of Mathematics - An invitation to effective thinking." by Edward B. Burger and Michael Starbird. Published by John Wiley & Sons. Inc. This book is available via Amazon.

First Assignment: Read in "What is Mathematics" Section 1 - Calculation with Integers, Section 2 - The infinitude of the number system - mathematical induction -- Parts 1, 2 and 4. Read Part 1 about prime numbers in the Supplement to Chapter 1. For homework, we have the following problems: (a) See if you can find a "pattern proof" (as we did in class for 1 + 3 + 5 + ... + (2n-1) = n x n) for the formula 1^3 + 2^3 + ... + n^3 = (1 + 2 + 3 + ... + n)^2 = (T_n)^2. You can work at this on your own or do some web browsing. Try searching google with the phrase "squared triangular number" and see what you get. Squared Triangular Number. Or search on google image for "proofs without words" and then explore what you get. Proofs Without Words. (b) Is 13721731779 a prime number? What about 13721731777? (c) Figure out good tests to determine if a number is divisible by 4, 6 or 8. (d) List all the prime numbers that are less than 100. Explain what you would do if you were asked to find all the prime numbers less than 500. The problems in this assignment are due Wednesday, August 31,2016.

Sixth Assignment: (You can do parts (A) and (B) immediately. Part (C) will need the new version of the Course Notes. THEY ARE HERE! SEE COURSE NOTES 2 ABOVE. (A) View Ulam Spiral 3. and make your own Ulam spiral starting at 41. Work out for yourself why it is that the Northeast and Southwest corners of the spiral are labeled by the numbers N^2 - N + 41. Suppose that you were to start the spiral at the number 17. Draw it! What would the number sequence be for the Northeast and Southwest corners of this spiral? Are there a lot of prime numbers in this sequence? (B) Here is a remarkable algebraic identity: (x^2 - y^2)^2 + (2xy)^2 = (x^2 +y^2)^2. Verify this by multiplying out and adding up the algebra on the left hand side and showing that it equals the algebra on the right hand side. Now try some examples of this identity by choosing integer values for x and y. For example, if x = 2 and y = 1, then we have (2^2 - 1^2)^2 + (2)^2 = (2^2 + 1^2)^2 and this is same as 3^2 + 4^2 = 5^2. So you see that our identity produces triples of numbers A,B,C so that A^2 + B^2 = C^2. Since these would be integer sides of a right triangle, such triplets are called Pythagorean Triplets after the Pythagorean Theorem. Find values of x and y that produce the triplet 5, 12, 13. Find other triplets by trying other values of x and y. Look up the Wikipedia article on the Pythagorean Theorem and find some part of it that you find interesting. Report on that part that you found. (C) Find integers a and b so that 73 a + 51 b = 1. Do this by using the Euclidean algorithm and then by using the matrix method that we discussed in class. (Notes on the matrix method will be available shortly.) Remember that you can check the continued fraction for 73/51 from the calculator on our website. You should find that 73/51 = [1,2,3,7]. Calculate the fraction for [7,3,2,1] = 7 + 1/(3 + 1/(2 + 1/1)) = P/Q: that is find the values of P and Q. You should find that P = 73 and 51Q == 1 mod(73). Next week we will show that this always happens! The problems in this assignment are due Wednesday, October 5, 2016.

Eighth Assignment: Read carefully the proof of Fermat's Theorem (pages 37-38 in "What is Mathematics"). We will discuss the proof in the class. (A) Not only is Fermat's Theorem true, but the following fact is also true: Let p be a prime number and let R(p) = {1,2,...,p-1} be the non-zero residue classes modulo p. Then then there is a number g in the set {1,2,...,p-1} such that the residues of g^1, g^2, g^3, g^4,..., g^(p-1) are exactly the set R(p). For example, let p = 5. and let g = 2. Then (using congruence mod 5) g^1 == 2, g^2 == 4, g^3 == 3, g^4== 1. Thus we get {1,2,3,4} in the order {2,4,3,1} from the powers of g. This problem asks you to find such a g when p = 13. (B) We have remarked in class then if P/Q = [a1,a2,...,an] where n is odd, and the sequence of numbers {a1,a2,...,an} is symmetric in the sense that it is the same as the reverse order {an,...,a2,a1}, then Q^2 == 1 mod P. For example [2,3,2] = 16/7 and 7^2 = 49 == 1 mod 16 (since 3 x 16 = 48). Find all residues g in {1,2,3,4,5,6,7,8,9,10,11} modulo 12 such that g^2 == 1 mod 12. Find the continued fraction expansion for each fraction 12/g where g^2 == 1 mod 12. Which of these fractions are symmetric? (C) Let p be a prime number. Suppose that x is a residue mod p in the set {1,2,3,...,p-1} and that x^2 == 1 mod p. Then we have that x^2 -1 == 0 mod p (by subtracting 1 from both sides of the equation). Since x^2 - 1 = (x+1)(x-1), we have (x+1)(x-1) == 0 mod p. Explain that this means that p divides (x-1)(x+1) and use the result that "if p divides ab then p divides a or p divides b", to conclude that p divides (x-1) or p divides (x+1). Now remember that p is prime and that x is in the set {1,2,3,...,p-1}, and show that the only x that can satisfy one of the other of these conditions is x = 1 or x = (p-1). Illustrate your reasoning using the prime p = 13. (D) Read Matrices and Complex Numbers, and do ALL the exercises in these pages. (E) After you have done (D) note that (x+iy)^2 = (x^2 - y^2) + i(2xy). What does this have to do with our Pythagorean formula (x^2 - y^2)^2 + (2xy)^2 = (x^2 +y^2)^2 ? (F) Examine the Mathematical Time Line. Choose a mathematical topic that occurred after 1600 and look it up. In your looking you can start by using a link provided by TimeLine, but then read that article or look further into references in the article. Keep reading until you find something of interest to you. Report on what you found. The problems in this assignment are due Wednesday, October 19, 2016.

Ninth Assignment: Do the problems in Five Problems. The problems in this assignment are postponed to, November 2, 2016. The next assignment contains an extension of this problem set with three new problems to be handed in on November 2, 2016.

Tenth Assignment: Do the problems in Five Plus Three Problems. See also Course Notes 2. The problems in this assignment are due Wednesday, November 2, 2016.

Twelveth Assignment: Do the problems in Problemset. Use Number Theory Notes 3. For the Tower of Hanoi, you may wish to use Tower of Hanoi Demonstrator. Look at the knot game at Ayaka Shimizu's Game. Play the game! Read Ayaka Shimizu's Paper about the game. Write about your experience playing this game. Can you find a good strategy for winning? The problems in this assignment are due Wednesday, November 16,2016.

Fourteenth Assignment: In this course, we have discussed a lot of different topics in mathematics, some that have "no lower age limit" and some that would be suitable for older children like ourselves. Make a list of as many "no lower age limit required" topics that you can think of (mathematical, but not limited to this course). Choose three such topics and describe how you would use them to arouse interest in mathematics in young learners. This assignment, is due on November 30, 2016.

BEYOND THIS POINT ARE MANY SUGGESTIONS FOR READING AND PROJECTS.

See UMG. This is the website for the Undergraduate Mathematics Journal. It a good source of interesting articles. See Old Hats for a good example of a paper published in the Journal.

See Conway's Army for a short article explaining a solitaire game and a proof about its limitations.

See Existence for an existence proof and a story related to the question whether existence proofs should exist.

See Geometry This is a summary of some remarks about geometry.

See ContFrac. This is a calculator on the web that converts numbers to their continued fractions. Try it on sqrt(2), sqrt(3), e, pi and other favorite numbers.

See VanPoorten. This is an excellent article on continued fractions.

See Recreations. This is an access page to programs and mathematical recreations.

See Proof This is a remark about proofs in mathematics.

See The Library of Babel for a short story by Jorge Luis Borges.

Recommended Reading: The Feeling of Power. This is a short story by Isaac Asimov.

Recommended Reading: InfiniteHotel. A cartoon story about an infinite hotel that can seemingly accomodate any new influx of guests.

See Boolean Notation. A short article about Boolean Algebra notation.

See Boolean Algebra. A short article about Boolean Algebra and some ideas related to it, including switching circuits, Laws of Form, recursions and circuits that count.

See Even/Odd. An article by David Joyce about Even and Odd Numbers.

See Peano Axioms for a list of the Peano axioms for the natural numbers and a list of problems to prove, using these axioms. Compare these axioms with the discussion in the article by David Joyce on Even and Odd.