Class Diary

For the homework problems listed below, the ones marked with * should be turned in and graded. The ones marked with ** are extra credit problems. The others are non-graded problems.

• Monday, August 25 -
  • Class: The 6174 magic trick, handed out syllabus, introduced the class, and started Section 1.1.
  • Read: Section 1.1 and 1.2 for Wednesday.
  • Do: No homework on the first day.
• Wednesday, August 27 -
  • Class: The division algorithm, Euclid's proof of the infinitude of the primes, the largest known prime and primality testing.
  • Read: Section 1.3 for Friday.
  • Do: 1(acdf), 3(af), 6, 7*, 9**, 10*, 11(cd), 12*, 14(a), 16(ab).
• Friday, August 29 -
  • Class: Review of induction, the distribution of primes, the prime number theorem, the Riemann hypothesis, Fermat primes, and the start of Section 1.3.
  • Read: Section 1.4 for class next Wednesday.
  • Do: 17, 21(a), 24*, 27* (see the hint for exercise 9), 30*, this problem**, and vote on which days you want the exam to occur (see the homework assignment posting for more details on this)*.
• Wednesday, September 3 -
  • Class: Discussed the homework, proved Prop 1.10 and Prop 1.11. Proved Lemma 1.12 and discussed the Euclidean algorithm.
  • Read: Section 1.5 for class on Friday.
  • Do: 32(abf), 33(ac), 36, 43*, 44*.
• Friday, September 5 -
  • Class: The fundamental theorem of arithmetic, computing GCDs and LCMs.
  • Read: Section 2.1 for class on Monday.
  • Do: 54(ac)*, 55, 56*, 59(acf), 60(cd), 64*.
• Monday, September 8 -
  • Class: Conclusion of Chapter 1, start of Section 2.1 (up through Prop 2.4).
  • Read: Section 2.2 for class on Wednesday.
  • Do: From Chapter 1: 76*, 77**, 78**, 87*. From Chapter 2: 1(ab), 3(b), 4(aef), 11*, 13*.
• Wednesday, September 10 -
  • Class: Finished Section 2.1, and discussed divisibility tests and day of the week magic trick.
  • Read: Nothing new.
  • Do: this problem, 18*, 20*, and 26*.
• Friday, September 12 -
  • Class: Section 2.2 - Theorem 2.6, computational techniques, and multiplicative inverses.
  • Read: Section 2.3 for Monday.
  • Do: 28(ad), 29(bd)*, 31**.
• Monday, September 15 -
  • Class: An application of Theorem 2.6 to ``dividing the coconuts,'' and an introduction, statement, and proof of the Chinese remainder theorem. I will give examples of the Chinese remainder theorem next time.
  • Read: Section 2.4 for Wednesday.
  • Do: 33(ac), 33(d)*, 34(b)*, 36*, this problem**.
• Wednesday, September 17 -
  • Class: An announcement about new Mersenne primes (see this page), an example of the Chinese remainder theorem, Wilson's theorem and converse, and the start of Section 2.5.
  • Read: Section 2.5 for Friday.
  • Do: 42(ac), 46* (assume that p is odd), and this problem*.
• Friday, September 19 -
  • Class: Another proof of Fermat's little theorem, applications, pseudoprimes, primality testing.
  • Read: Section 2.6 for Monday.
  • Do: 51(a), 51(d)*, 52, 58*, 60*, and this problem**.
• Monday, September 22 -
  • Class: Definition of the Euler phi function, and the proof of Euler's theorem.
  • Read: Nothing new for Wednesday.
  • Do: 66, 68(bc), 73*, 76*.
• Wednesday, September 24 -
  • Class: Irrationality of the square root of 2, and discussion of the decimal expansions of rational numbers. (This material corresponds roughly to Section 7.1 of the book).
  • Read: Nothing new for Friday.
  • Do: this problem, this problem*, and this problem*.
• Friday, September 26 -
  • Class: Discussion of cryptography and the RSA cryptosystem based on integer factorization (there is some discussion of this in Section 8.2 in the book).
  • Read: Section 3.1 for next Monday.
  • Do: this problem*, and this problem*. A text file containing the 308 digit number n can be found here.
• Monday, September 29 -
  • Class: Answered some homework questions, started Section 3.1, ending with the statement of Theorem 3.1.
  • Read: Section 3.2 for Wednesday.
  • Do: 2, 3*, 5(ab), 5(fg)*.
• Wednesday, October 1 -
  • Class: Distributed information about the exam (available here). Did Exercise 8 and proved Theorem 3.1.
  • Read: Nothing new.
  • Do: 7*, this problem*.
• Friday, October 3 -
  • Class: Covered Section 3.2 on Euler's phi function.
  • Read: Notes and homework problems from Chapters 1 and 2 in preparation for exam review next Monday.
  • Do: 10(acf), 13(ab), this problem, 19*, 20*, 25*, this problem**, this problem**.
• Monday, October 6 -
  • Class: Office hours and exam review.
  • Read: Materials to prepare for the exam on Wednesday.
  • Do: Study!
• Wednesday, October 8-
  • Class: Exam I.
  • Read: Something relaxing.
  • Do: Relax!
• Friday, October 10-
  • Class: Ceiling fans, the number of positive divisors, and the sum of divisors functions.
  • Read: Section 3.3 and 3.4, and Section 3.5 for next Monday.
  • Do: 30(fgh), 34*, 37*, 40**, this problem**, 42(cg), 43, 49*.
• Monday, October 13-
  • Class: Returned exams and went over them. Talked about projects.
  • Read: Project descriptions (given here).
  • Do: Pick a project!
• Wednesday, October 15-
  • Class: The Banach-Tarski paradox, perfect numbers, and the Mobius mu function.
  • Read: Section 3.6 for Friday.
  • Do: 56(adf)*, 60**, 61*.
• Friday, October 17-
  • Class: The Mobius inversion formula.
  • Read: Nothing new for Monday.
  • Do: 63*, 64*, 65*, this problem**, and this problem**.
• Monday, October 20-
  • Class: Roots of unity, an application of the Mobius inversion formula, cyclotomic polynomials.
  • Read: The material I covered today, and on Wednesday is not in the book. For material on cyclotomic polynomials, look at this page.
  • Do: this problem, this problem*, this problem**, and this problem**.
• Wednesday, October 22-
  • Class: Dirichlet series and the Riemann hypothesis.
  • Read: The material I covered today is not in the book. For material about Dirichlet series, see this page.
  • Do: this problem*, this problem*, and this problem**.
• Friday, October 24-
  • Class: Reviewed the four "important" arithmetic functions, covered section 4.1 on quadratic residues and quadratic congruences.
  • Read: Section 4.1 and 4.2.
  • Do: 4.1, 4.2, 4.6*, 4.7*, 4.10(b)*, and this problem**.
• Monday, October 27-
  • Class: Talked about cyclotomic polynomial homework. Stated an proved Euler's criterion, and gave applications (Proposition 4.5, and the infinitude of primes congruent to 1 mod 4).
  • Read: The proof of the infinitude of the primes. You can download it here. For a version in green, go here.
  • Do: 4.12(a)*, 4.14(ab), 4.18*, 4.22*.
• Wednesday, October 29-
  • Class: Recall election for second exam date, went over homework, introduced quadratic reciprocity.
  • Read: Nothing new.
  • Do: This problem*.
• Friday, October 31-
  • Class: Stated and proved Eisenstein's lemma, determined when 2 is a square mod p, and explained how to use quadratic reciprocity to compute Legendre symbols.
  • Read: Section 4.3.
  • Do: 4.28(bce)*, 4.30*, 4.35(b)*.
• Monday, November 3-
  • Class: Proved the law of quadratic reciprocity and talked about an application.
  • Read: The proof of the law of quadratic reciprocity. I've typed up notes of the proof and posted them here.
  • Do: 4.27*, this problem**.
• Wednesday, November 5-
  • Class: Covered Section 5.1 on primitive roots.
  • Read: Section 5.1 based on today's class, and Section 5.2 for Friday.
  • Do: 1(bd), 2(a), 3(c)*, 7*, 8*, this problem**.
• Friday, November 7-
  • Class: Proved that primitive roots exist for prime numbers.
  • Read: Look over Section 5.1 and 5.2 to clarify the concepts and see if you have any questions for Monday.
  • Do: 10(ac)*, 11(b)*, 12, 19, 21*. [These are the problems Melanie chose. I would have chosen the problems 10(ab), 11(b)*, 12(a)*, 18*].
• Monday, November 10-
  • Class: Answered questions, summarized material from Section 5.1 and 5.2, and gave applications.
  • Read: Section 5.3.
  • Do: 9*, this problem*, and this problem**.
• Wednesday, November 12-
  • Class: Magic trick and discussion of Benford's law.
  • Read: Look over notes from Chapters 3 and 4 and prepare questions to ask in class on Friday.
  • Do: Study for the exam.
• Friday, November 14-
  • Class: Student questions related to the exam.
  • Read: Notes and homework from Chapters 3 and 4.
  • Do: Study!
• Monday, November 17-
  • Class: Exam 2.
  • Read: Something relaxing.
  • Do: Take a break!
• Wednesday, November 19-
  • Class: Started Section 5.3. Proved one half of Theorem 5.19 (up through Corollary 5.13).
  • Read: Anything from Section 5.3 you're not clear on.
  • Do: 23*, 27*.
• Friday, November 21-
  • Class: The divisor game, proved the other half of Theorem 5.19.
  • Read: Something relaxing.
  • Do: 24(ab)*, 25(ab)*.
• Monday, December 1-
  • Class: Section 6.1, including applications to football and measuring liquid.
  • Read: Section 6.1, and Section 6.3 for Wednesday. We're skipping Section 6.2.
  • Do: 3, this problem*, this problem*, 10(c)**.
• Wednesday, December 3-
  • Class: Pythagorean triples and their classification (Section 6.3).
  • Read: Section 6.4.
  • Do: 13(a), 14(ab)*, 15*, 16(a)*.
• Friday, December 5-
  • Class: Fermat's Last Theorem (Section 6.4).
  • Read: Nothing else!
  • Do: 21*, this problem*.

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