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Homework assignments
(MATH 5/4161-001, Spring 2014)

Last update: Friday, April 18, 2014

Disclaimer: The information below comes with no warranty. If, due to typographical error, there is a discrepancy between the exercise numbers announced in class and the numbers below, or this page is not completely up to date, the required homework consists of those exercises which were announced in class. Check for the time of last update above. If, by my mistake, a wrong exercise number shows up below, I will allow you extra time to hand in the exercise whose number was announced in class. If, however, exercise numbers are missing because this page is not up to date, it is your responsibility to contact me before the due date. (No extra time will be allowed in that case.) This page is up to date if the last update happened after the last class before the next due date.

Notation: 1.2/1b means Exercise 1b in Section 1.2. The deadlines do not apply to the Bonus questions, which expire only once we solve them in class.

No. Date due: Problems:

13 Th 4/24 9.1/2,4,5a,5b,7; 9.2/4a, 4b (you may use 4a even if you do not succeed proving it), 5.

12 Th 4/17 8.3/2a,2b,6b; 8.4/5,11.
Additionally, 5000-level students are expected to hand in a proof of the fact that the multiplicative group of every finite field is cyclic.

11 Th 4/10 8.2/1a,1b,6a,8a; 10.1/1,3a,3c.

10 Th 4/3 8.1/2a,2c,6a,11a,11b.

9 Th 3/27 7.3/1a, 1c, 4,9 7.4/2, 5a, 5c, 8.
Our second test is on Tuesday March 25. You may download the Sample Test II to prepare for it.

8 Th 3/20 6.3/2c,5a; 7.2/2,3,6,8.
Bonus:

6.3/6a
Prove or disprove that Euler's phi-function φ satisfies φ(n²)=nφ(n) for all positive integer n.

7 Th 3/13 5.3/1a,1b,5a,5b; 6.1/7a,7b,8,9,10a; 6.2/1a,2,4a,4c,7b,8a (use 6.2/3).

6 Th 2/27 4.4/1a,1c,1d,5,10,17; 5.2/1,4a,4b.

5 Th 2/13 4.3/2b, 2c, 5a, 5b, 9; 4.2/2,6a,6c,10.
Our first test is on Tuesday February 11. You may download the Sample Test I to prepare for it.

4 Th 2/6 3.2/4a, 4b, 5; 4.2/5,8b,9.
Bonus question:

Let p be a prime and n be any positive integer. Let k be the largest exponent for which p^k divides n! (that is, n factorial). Express k in terms of n and the sum of digits of n, when n is written in base p. Prove your formula. Note added on Tuesday, March 11, 2014: this bonus question is now available for reduced credit, after seeing 6.3/7.

3 Th 1/30 2.4/1; 2.5/1 (also solve those Diophantine equations which can be solved); 3.1/3a, 3b, 6b, 6c.
Bonus questions:

Consider the Euclidean Algorithm presented in section 2.4. Express r₅ as a linear combination of a and b.
Generalize 1.2/2 to expressing the n^th row k^th entry in Pascal's triangle as a linear combination of entries in the (n-r)^th row, where r is an arbitrary positive integer.

2 Th 1/23 2.1/1a, 1c, 8; 2.2/1,2,3b,3c,11; 2.3/1,4b,14a.

1 Th 1/16 1.1/1b, 1c, 9; 1.2/2, 3d, 5a.

No.	Date due:	Problems:
13	Th 4/24	9.1/2,4,5a,5b,7; 9.2/4a, 4b (you may use 4a even if you do not succeed proving it), 5.
12	Th 4/17	8.3/2a,2b,6b; 8.4/5,11. Additionally, 5000-level students are expected to hand in a proof of the fact that the multiplicative group of every finite field is cyclic.
11	Th 4/10	8.2/1a,1b,6a,8a; 10.1/1,3a,3c.
10	Th 4/3	8.1/2a,2c,6a,11a,11b.
9	Th 3/27	7.3/1a, 1c, 4,9 7.4/2, 5a, 5c, 8. Our second test is on Tuesday March 25. You may download the Sample Test II to prepare for it.
8	Th 3/20	6.3/2c,5a; 7.2/2,3,6,8. Bonus: 6.3/6a Prove or disprove that Euler's phi-function φ satisfies φ(n²)=nφ(n) for all positive integer n.
7	Th 3/13	5.3/1a,1b,5a,5b; 6.1/7a,7b,8,9,10a; 6.2/1a,2,4a,4c,7b,8a (use 6.2/3).
6	Th 2/27	4.4/1a,1c,1d,5,10,17; 5.2/1,4a,4b.
5	Th 2/13	4.3/2b, 2c, 5a, 5b, 9; 4.2/2,6a,6c,10. Our first test is on Tuesday February 11. You may download the Sample Test I to prepare for it.
4	Th 2/6	3.2/4a, 4b, 5; 4.2/5,8b,9. Bonus question: Let p be a prime and n be any positive integer. Let k be the largest exponent for which p^k divides n! (that is, n factorial). Express k in terms of n and the sum of digits of n, when n is written in base p. Prove your formula. Note added on Tuesday, March 11, 2014: this bonus question is now available for reduced credit, after seeing 6.3/7.
3	Th 1/30	2.4/1; 2.5/1 (also solve those Diophantine equations which can be solved); 3.1/3a, 3b, 6b, 6c. Bonus questions: Consider the Euclidean Algorithm presented in section 2.4. Express r₅ as a linear combination of a and b. Generalize 1.2/2 to expressing the n^th row k^th entry in Pascal's triangle as a linear combination of entries in the (n-r)^th row, where r is an arbitrary positive integer.
2	Th 1/23	2.1/1a, 1c, 8; 2.2/1,2,3b,3c,11; 2.3/1,4b,14a.
1	Th 1/16	1.1/1b, 1c, 9; 1.2/2, 3d, 5a.