Collaborative Homework Assignments:

These are additional homework problems you should work on.  Feel free to talk about these problems with your classmates, as well as your instructors.  We will be looking at some of these problems almost every day, expecting that students will present solutions in class.  They are listed below by section number, rather than by day.

1.1:  1, 6

1.2:  3a-d, 10

1.3:  2, 3

2.1:  5, 10

2.2:  5, 14

2.3:  1, 2ab, 4cd

2.4:  1, 2a, 3b, 6

3.1:  3, 8

3.2:  1, 8

4.2:  1, 2, 4, 6a

4.3:   3, 6ab

4.4:  1abcd, 2abc, 5, 10

5.3:  1, 2a, 5, 11a, 16a

5.4:  9, 11

7.2:  7

7.3:  8

8.1:  2a, 4, 12b

Consider the prime 19.  Write down the possible orders for integers modulo 19.  Now, it turns out that 2 is a primitive root for this prime.  Show this explicitly, by computing all the powers of 2 to get the integers 1 through 18, recording your results in a table.  Then use our theorem to compute the orders of all these integers; record the orders in your table too (of course, we hope that the orders we get are on your list of possibilities).  Now make a new table, recording the number of elements with these orders, below the possibilities.  Finally, enter a new row with the values of the Euler-phi function.  The last two rows should be the same!!!

8.2:  3, 6

Take your data from 19, and determine which elements modulo 19 are quadratic residues.  Express each quadratic residue as the square of two different integers chosen from 1 through 19.  Check that Euler's criterion works for each of the non-zero residues.

9.1: 1, 4, 11b

9.2: 1, 11

9.3:  1, 3, 4

11.1:  1, 4

11.2:  1