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Highlighted homework is due soon |
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Date |
Assignment |
| Material for Test 2 ends here (Chapter 37) | |
| 29 July |
Use case analysis to show that if \(|x+y|\leq|a|\), then \(|x|\leq|a|+|y|\). Show that 3 is prime in \(\mathbb{Z}[\,i\sqrt 5\,]\) — use the same strategy I used to show 2 is prime in \(\mathbb{Z}[\,i\sqrt{5}\,]\) 35.3, 35.4, 35.6, 36.2, 36.3, 36.4, 36.5, 37.1, 37.2, 37.3 (need \(i\) for (d)), 37.5, 37.6, 37.8 Extra Credit: 35.9 |
| 22 July |
16.1, 16.3, 17.1, 17.2, 17.4, 18.1 (started in class), 18.2 Divide (i.e., find quotient and remainder) the Gaussian integers:
17.4 Factor out the gcd, solve the new problem, use it to construct a solution to the original.
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| 20 July | Read Chapter 37 |
| 15 July |
Read Chapters 35, 36 14., 14.2 The extra credit mentioned in class: if \(\gcd(m,n)=1\) and \(x\) divides \(mn\) and \(x\equiv_m a\) and \(x\equiv_n b\) then \(a\mid m\) and \(b\mid n\). (Note: The statement is false! Find a counterexample instead.) 15.2, 15.3, 15.5, 15.6 Hints:
14.1 Consider the case where \(n\) is odd, so that \(x^n+1=(x+1)(x^{n-1}-x^{n-2}+x^{n-3}-x^{n-4}+\cdots-x+1)\), then the case where \(n\) is even, but not a power of 2, in which case \(n=q\cdot 2^k\), where \(q\) is odd.
14.2 Try an example, say \(m=3\) and \(k=4\), then \(m=3\) and \(k=5\). Look for a pattern to the division of \(F_{k-2}\) by \(F_m\). Then show that, for any odd number, \(\gcd(a,a-2)=1\). |
| 13 July | Read Chapter 18 |
| 8 July | Read Chapter 16 Prove that any prime number larger than 4 is congruent to 1 or 5 modulo 6 11.5, 11.6, 12.2, 13.3, (for graduates: 13.6) Extra Credit: 12.6, 13.2, (for undergraduates:) 13.6 |
| 6 July | Read Chapter 14 |
| Material for Test 1 ends here | |
| 1 July | 9.1, 9.3, 10.1, 10.2, 11.1, 11.2, 11.3, 11.10 |
| 24 Jun | Read Ch. 9 7.1, 7.2, 7.3, 8.2, 8.3, 8.4 (b)–(e), 8.5, 8.9 |
| 17 Jun | Read Ch. 7 5.1, 5.3, 5.4, 6.1, 6.2, 6.5, 6.6 |
| 10 Jun | 2.2, 2.6, 2.7, 3.1, 3.2, 3.3 |
| 3 Jun | 1.1, 1.2, 1.6 |