- MNT 7.2
- MNT 7.3
- MNT 7.16
- MNT 7.18
- Let F3(i) be the field with 9 elements, where i is an element whose square is -1. Find all the generators of the group of units of F3(i).
- Find all the generators of the group of units of Z/11Z.

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# Math 1020 Applied Elementary Number Theory

## University of Pittsburgh Math Department

# HW 6 due Wed October 12, 2016

# A pseudo-random number generator

# HW5, due Wed Oct 5, 2016

# Midterm Exam, Wednesay, Oct 19, 2016

# Two proofs of Quadratic Reciprocity

# HW4, due Wed Sep 28, 2016

# HW3, due Wednesday Sep 21, 2016

- MNT 7.2
- MNT 7.3
- MNT 7.16
- MNT 7.18
- Let F3(i) be the field with 9 elements, where i is an element whose square is -1. Find all the generators of the group of units of F3(i).
- Find all the generators of the group of units of Z/11Z.

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Here is a link to the Blum-Blum-Shub paper.

Study the proof of quadratic reciprocity in MNT, as well as Rousseau’s proof based on the Chinese Remainder theorem.

- MNT Problem 5.7
- MNT Problem 5.9
- MNT Problem 5.12
- MNT Problem 5.17
- MNT Problem 5.22
- MNT Problem 5.38 (first part, but not the generalization)

There will be an midterm exam on Wednesday, October 19, 2016 during our regularly scheduled class.

I recommend two particular proofs of quadratic reciprocity.

Exercises Chapter 3, MNT page 37

- problem 3.5,
- problem 3.10,
- problem 3.14
- problem 3.16

Exercises Chapter 4, MNT page 49

- optional/bonus problem 4.22

Exercises Chapter 5, MNT page 63-65

- problem 5.3
- problem 5.16.

Find all the quadratic residues modulo 41. You can use WolframAlpha or other computer programs to find the answer. In WolframAlpha Mod[x,41] computes x modulo 41.

Chapter 2, MNT pages 26-27

- Problem 11
- Problem 13 (Hint: insert a “k” where appropriate in the argument on page 19).
- Problem 15a (Hint: do you recognize this as a Dirichlet product? How is the formula related to the Moebius inversion formula?)
- Problem 26a,b. There is no need to prove convergence of either side. For full credit, it is enough to work the case s=2. (Hint: use the product formula over primes in problem 25 for zeta(s).)

Chapter 3, MNT pages 36-37

- Problem 2
- Problem 4 (Hint: if an equation has no solutions modulo m for some m, then it has no solutions in the integers.)