Math 347 Honors, Spring 2009

Regular homework assignments

1. DUE 1-30-09: 1.4, 1.7, 1.20, 1.27, 1.28. BONUS 1.26 (Hint: work #25 first).
2. DUE 2-6-09 : 1.15, 1.38, 1.39, 1.41 (d), 1.47 (a), 1.49 (b,c,d,e). BONUS 1.47 (b)
3. DUE 2-13-09 : 2.4, 2.10 (a-e), 2.21, 2.22, 2.23, 2.24, 2.31, 2.32. Proofs are required only for 2.21, 2.24, 2.32.
4. DUE 2-20-09 : 2.19, 2.38, 2.47, 3.5, 3.16, 3.18. proofs required for all problems except 2.19.
5. DUE 2-27-09 : 3.4, 3.24, 3.31, 3.41, 3.49 (b), 3.56. BONUS 3.59.
6. DUE 3-9-09 (Monday) : 4.11, 4.12 (a,b,c), 4.24, 4.25 (c,d,e), 4.26, 4.29 (a,b), 4.31, 4.34
7. DUE 3-13-09 : 5.5, 5.10, 5.12, 5.19 (k=2,3 only), 5.28, 5.42, 5.45
8. DUE 3-20-09 : 6.4, 6.17, 6.18, 6.22, 6.24, 6.28, 6.29
9. DUE 4-3-09 : 6.31 (b), 6.40, 6.50 (a), 7.6, 7.11, 7.14, 7.18. BONUS: Show that for every composite n>4, n|(n-1)!
10. DUE 4-10-09 : 7.24, 7.30, 10.4, 10.7, 10.9, 10.14, 10.17
11. DUE 4-22-09 : 10.25, 10.29 (replace 200 with 20,000), 13.8, 13.10, 13.11, 13.22. BONUS 10.32
12. DUE 4-27-09 : 13.20, 13.23, 13.25, 13.26, 13.29, 13.30
13. DUE 5-4-09 : 14.3, 14.8, 14.12, 14.13, 14.15, 14.23, 14.33. BONUS 14.55

Honors Homework assignments

1. DUE Wednesday, March 4.
   (a) Prove the trianlge inequality without using any operations "squaring" or "taking square roots". Hint: consider different cases, e.g. x and y both positive, etc.
   (b) 1.31 (a), 1.55, 1.56
   (c) 2.40
   (d) 3.54, 3.57, 3.58
   

2. DUE Wednesday, April 15.
   (a) 4.49, 4.51
   (b) 5.46, 5.52
   (c) For positive integer n, let s be the binomial coefficient (2n choose n). Show that s is divisible by every prime in [n+1,2n]. Conclude that the number of primes in (n,2n] is at most 2n/(log₂ n), where (log₂ n) is the logarithm base 2 of n. Next, show that the number of primes in [1,2^k] is at most 1+ (sum from m=2 to k) 2^m/(m-1). Divide the sum into terms with m < k/2 and m &ge k/2, and show that the number of primes is at most 1 + (4/k) 2^k + 2^2+k/2. Finally, show that for any x>2, the number of primes in [1,x] is at most 20 x/(log₂ x).
   

Homework Solutions

Exam #1. Friday, March 6. Covers Chapters 1, 2, 3.