MAT 421 Suggested HW Problems

Highlighted homework is due soon

Date	Assignment
6 May	Some study questions: Find a primitive number modulo 17. Why is 10403 a bad modulus to use in RSA encryption? Why is the factorization of \(2=(1+i)(1-i)\) considered legitimate in \(\mathbb Z[i]\), but \(1+i=i(1-i)\) is considered trivial? Does 3 factor in \(\mathbb Z[i]\)? Does 5? Exercises 35.5, 35.6, 35.7, 35.8 Show that \(\sqrt p\) is irrational for any prime number \(p\). Show that \(\sqrt m\) is irrational for any integer \(m\) that is not a perfect square. Show that \(\root n\of p\) is irrational for any natural \(n>1\). Exercises 37.5, 37.6, 37.7, 37.8
1 May	Read Chapter 18, if you haven’t already… Complete the assignment at this link (PDF file) Read Chapters 35, 36, 37
17 Apr	Exercises 17.1, 17.2, 17.4
10 Apr	Exercises 15.2, 15.3, 15.5, 15.6 Read Chapter 16, especially noting what a “Carmichael number” is Exercises 16.1, 16.3 Read Chapter 18
4 Apr	Exercises 11.1, 11.2, 11.3, 11.5, 11.12, 12.2, 13.3 (note: I deleted some problems from Chapter 11) Choose two moduli that allow you to solve \[x\equiv a\pmod m\\ x\equiv b\pmod n\] with a unique \(x\) modulo 60 (as discussed in class on 27 Mar 2019). Take the last two digits of your USM ID, and call that number \(z\). Choose \(a,b\in\lbrace1,2,\ldots,60\rbrace\) such that \(a\equiv z\pmod m\) and \(b\equiv z\pmod n\). Then solve \[x\equiv a\pmod m\\x\equiv b\pmod n\] and verify that \(x\equiv z\pmod{60}\).
2 Apr	Read Chapter 14 (there will probably be a quiz)
6 Mar	Exercises 8.2, 8.3, 8.4(a,b,e), 8.5(b,c), 8.9 Exercises 9.1, 9.2 Exercise 10.2
20 Feb	Exercises 7.1, 7.2, 7.6
18 Feb	Read Chapters 7 (especially the \(\mathbb E\)-Zone) and 8
♥♥♥14 Feb♥♥♥	Handout on ♥♥♥Fermat’s Last Theorem♥♥♥
♥♥♥1314 Feb♥♥♥	Read Chapters 2, 3, 4 and then do Exercises 2.2, 2.6, 2.7, 3.1, 3.2, 3.3
♥♥♥1114 Feb♥♥♥	Exercises 5.1, 5.3, 5.4, 6.1, 6.2, 6.5 Bonus: 6.6 (this is kind of tough)
4 Feb	Exercises 1.1, 1.2, 1.6 Also, Read Chapter 6
30 Jan	Read Chapter 4