HOMEWORK: click here for solutions (DO NOT look at these solutions until you have either turned in the HW, or are done working on it).
Please spend at least an hour a day reading the material / looking at the proofs / making sure you understand
the details. Below is a tentative reading list and homework assignments. It is
subject to changes depending on the amount of material covered each
week. I strongly encourage you to skim the reading before class, so you are
familiar with the definitions, concepts, and the statements of the material
we'll cover that day. Also, you can frequently do the homework well before it is due -- you are urged to start working on the problems as soon as you can.
First Unit: From the Fibonacci Numbers to Generating Functions
- Week 1: September 5 to 9
- Read Chapter 37 (Fibonacci's rabbits...).
- Homework (due Thursday, September 15): Page 305: #37.1abce (note gcd means greatest common divisor), #37.3, #37.5a (only do 5 terms), #37.9.
- Extra Credit (due Thursday, September 15): Page 305: #37.2a.
- Week 2: September 12 to 16
- Read: Chapter 36 (Binomial coefficients...) and Chapter 41 (Generating functions). Chapter 1, Chapter 2, Chapter 3.
- Homework (due Tuesday, September 20): #36.3, #36.6a, Also prove by direct computation that (n choose k-1) + (n choose k) = (n+1 choose k). #41.9ab.
- Extra Credit: #41/1a, #41.5 (hint: a_n = (n+1)n/2), #41.8 (see Theorem 11.1, page 72).
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Week 3: September 19 to 23
- Read: Chapters 1, 2 and 3 and Chapter 5. For more on the Euclidean Algorithm, see Chapter 1 of my book. I'll cover efficient algorithms (Section 1.2), which includes the Euclidean Algorithm.
- Homework (due Tuesday, September 27): Chapter 1, page 11: #1.2, #1.3, #1.4. Chapter 2, page 18: #2.1a, #2.5a (just do b = 4T_5). Also, prove that if ax^2 + bx + c = 0 has one rational root and a,b,c are integers then the other root is rational.
- Extra Credit (due Tuesday, September 27): Chapter 1, page 11: #1.1. Chapter 2, page 18: #2.4, #2.5b.
- Extra Credit (hard): Without using other sources, write down the generating function for the Catalan numbers and find the formula for the n-th Catalan number. (If you have not seen the Catalan numbers in a combinatorics class, you may google or wikipedia them, but do not read the proof of the generating function).
- Week 4: Sept 26 to 30
- Read: Chapters 5, 6 and 7 and my notes on the Euclidean Algorithm, see Chapter 1 of my book. I'll cover efficient algorithms (Section 1.2), which includes the Euclidean Algorithm.
- Read Chapter 1 of my book (up to the bottom of page 21) for an introduction to group theory, cryptography, modular arithmetic and RSA.
- Homework (due Tuesday, October 4): Chapter 3, page 23: #3.2, #3.3. Chapter 5, page 32: #5.1a, #5.3, #5.4abce
- Extra Credit: Read about the 3n+1 algorithm (Chapter 5, page 33, #5.5). Prove that there are infinitely many numbers which iterate to 1 under this map.
- Week 5: Oct 3 to 7
- Read: Chapter 1 of my book. Also from Silverman: Chapters 8, 9, 10.
- Homework (due Thursday, October 11): From Chapter 1 of my book: Page 4, #1.1.1. Page 9, #1.2.17. Page 15: #1.3.9, #1.3.10 (but do not do the last part asking to generalize). Also do the following (modified from Silverman's book): (1) Look at all numbers of the form 3x + 5y, where x and y are non-negative integers. What values occur as we vary x and y? Prove your conjecture. (2) Let x, y, and z range over the positive integers. Describe all numbers of the form 6x + 10y + 15z. In general, if a, b and c are given and x, y and z range over all positive integers, what numbers are attainable as ax + by + cz? (3) Gather data and make some conjectures (prove if possible): let x and y be relatively prime positive integers. Are there non-zero integers a and b such that a^2 x + b^2 y = 1? What about a^3 x + b^3 y = 1? Maybe the answer is yes for some x and y and no for others....
- Extra Credit (due Tuesday, October 11): Chapter 1 of my book: Page 14, #1.2.34 and #1.2.35.
- Week 6: October 10 to 14 (no class Tuesday, October 11)
- Read: Chapters 12 and 13 from Silverman, and Chapter 1 from "Proofs from THE BOOK" (it's okay if not all of the proofs are accessible now).
- Homework: Due Tuesday, October 18: None.
- Extra Credit: Chapter 13: #13.6.
- Week 7: October 17 to 21
- Read: Chapter 14.
- Will go over LaTeX on Tuesday the 18th, as well as discussing old HW problems and efficient proofs.
- Homework: Due Tuesday, October 25: Chapter 12, Page 83: #12.2a, #12.5abc. Chapter 13, Page 89: #13.1b, #13.3 (use this to conclude that there are arbitrarily large gaps between primes, or, in other words, we can have any finite number of consecutive numbers be composite), #13.5. Also (1) Prove Sum_{n = 1 to oo} 1/n^2 converges, and find its value to within 1/2011.
- Midterm replacement: Find the error in this proof of Goldbach and that counts as full credit for 3 problems on the midterm or final! (OK, to be fair, you can also prove it's correct!)
- Week 8: October 24 to 28 (No class Tues, Nov 2nd for extra time for exam / project)
- Week 9: Oct 31 to Nov 4: No class on Tuesday to work on exam / projects
- Week 10: November 7 to 11: Remember no class on Thursday
- Mathematica program on equidistribution and differences.
- Read: TBD (let me know your vote for the next unit).
- Thursday class: there will be no lecture on Thursday; I would like to meet with everyone for 5 or 10 minutes on Thursday. Let me know if you can do a meeting before class. I want to go over your exams with you individually.
- Homework: Due Tuesday, November 15: Chapter 5 of my book. Exercise 4.24 (just do the first part, for p/q to be a root). Also do: wait at least a day from reading any notes or solution keys, and redo any one problem from the midterm. Write it up cleanly, concisely and correctly.
- Extra Credit: Chapter 5 of my book. Exercise 7.8.
- Week 11: November 14 to 18:
- Mathematica program on Carmichael numbers.
- Read: Chapter 19 of Silverman's book
- Homework: Dues Tuesday, November 22: Silverman's book: Chapter 19: #19.3abc, #19.4a (as well as find one number that works). Also state a result about Carmichael numbers that is not covered in Chapter 19 of the book, or the three papers mentioned above. In other words, do some web searching to find something.
- Extra Credit: Formulate a conjecture about Carmichael numbers that has a chance of being true, and do a little web searching to see if anyone has investigated it.
If you want, please do the first few problems from the final by start of
class on Tuesday, December 6th.