MATH 116: Material Covered in Class and Quiz/Exam Information

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1 MATH 116: Material Covered in Class and Quiz/Exam Information August 23 rd. Syllabus. Divisibility and linear combinations. Example 1: Proof of Theorem 2.4 parts (a), (c), and (g). Example 2: Exercise 2.7 parts (a) and (b). Example 3: Prove that if a b and b c then a c. Example 4: Exercise 2.8 parts (a) and (b). Example 5: Exercise 2.14 parts (a) and (b). August 28 th. gcd(a, b) and Worksheet. Example 6: Discussion on Exercise 2.14 with 5 2. Example 7: gcd(6, 9) = gcd( 3, 9) = 3. Example 8: Lemma Example 9: Exercise August 30 th. Prime numbers. Quiz 1 (up to 2.17). Example 10: Exercise Example 11: Using Lemma 2.43, prove Lemma Example 12: Theorem Example 13: Use Theorem 2.49 to get the sieve of Eratosthenes. Example 14: After rambling a bit about prime distribution we solved Exercise September 6 th. Prime factorization and gcd (again). Quiz 2 (up to 2.58). Example 15: Exercise Example 16: Exercise Example 17: Find gcd( , ) Example 18: Various examples of squares and/or square-free numbers. Example 19: Exercise Example 20: Exercise 3.4. Example 21: Exercise 3.6. September 11 th. The Euclidean algorithm and Worksheet. Example 22: Exercise 3.7. Example 23: gcd(3172, 793) and gcd(3175, 793) Example 24: Lemma Example 25: Exercise

2 September 13 th. Bézout s Lemma. Quiz 3 (up to 3.22). Example 26: Exercise 3.32(f). Example 27: Find s and t in Bézout s Lemma for the numbers in Exercise 3.32(f). Example 28: Exercise Example 29: Exercise Example 30: Exercise September 18 th. Diophantine Equations and Worksheet. Example 31: Lemma 4.7 Example 32: Find solutions of 3x 9y = 12 in Z. Example 33: Find solutions of 2017x + 31y = 4 in Z. September 20 th. 8:00-8:50 Missing Proofs. 8:50-9:50 Exam 1 (Chapters 2 and 3). Example 34: Proof of Bézout s Lemma. Example 35: Proof of Theorem 4.9. September 25 th. Diophantine Equations, Z m, and Worksheet. Example 36: Exercise 4.11 (c). Example 37: Exercise Example 38: We found all equivalence classes modulo 5. September 27 th. Z m. Quiz 4 (Linear Diophantine equations). Example 39: Compute the following in Z 78 : Example 40: Exercise 6.23, parts (a) and (b). Example 41: Exercise 6.30(a). October 2 nd. More computations modulo m. Congruence equations. Example 42: Exercise Example 43: Exercise Not complete, though. Example 44: Exercise 7.10 (e). October 4 th. Congruence equations. Chinese remainder theorem. Quiz 5 (Chapter 5, and Chapter 6 up to Exercise 6.23). Example 45: Theorem 7.5. Example 46: Theorem 7.8. Example 47: Exercise 7.10 parts (a) and (f). 2

3 Example 48: Solve 2x 3 (mod 5). October 9 th. The Chinese remainder theorem. Worksheet. Example 49: Solve in two different ways. x 1 (mod 2) x 2 (mod 5) x 3 (mod 7) October 11 th. Z m. Exam 2 (Chapters 4 6, and Chapter 7 up to Section 7.2, included). Example 50: Find all the elements in Z 8, Z 10, and Z p for p prime. Example 51: Theorem Example 52: Compute φ( ) October 16 th. Properties of the Euler-φ function. Worksheet. Example 53: Theorem Example 54: Theorem 8.21 for k = 1 and k = 2. October 18 th. Euler s Theorem. Order of an element. Quiz 6 (Chinese remainder theorem and basic properties of Z m and φ). Example 55: Exercise 8.21 for k = 4. Example 56: is divisible by 7. Example 57: Prove that if gcd(a, b) = 1 then a φ(b) + b φ(a) 1 (mod ab). October 23 th. Order of an element. Worksheet. Example 58: Theorem 9.5 and Exercise 9.6. Example 59: Corollary 9.7. Example 60: Find Example 61: Exercise 9.149a). October 25 th. Primitive elements. Quiz 7 (Euler s φ-function and Euler s theorem). Example 62: Exercise 9.16(a). Example 63: Found primitive roots of n = 2, 3, 23, and 9. Checked that n = 8 and n = 12 do not have primitive roots. October 30 th. Existence of primitive roots. Example 64: Exercise

4 Example 65: Theorem Example 66: Exercise While solving this we looked at Corollaries 9.35 and November 1 st. Discrete logarithms. Solving equations using logarithms. Worksheet/Quiz. Example 67: Solve 7x 7 3 (mod 11). Example 68: Solve 21x (mod 38). November 6 th. Missing proofs and squares in Z n. Example 69: Theorem Example 70: Theorem Example 71: If r is a primitive root of p (odd prime) then r (p 1)/2 1 (mod p). November 8 th. Squares in Z n. Exam 3 (Section 7.3 to Chapter 9, both included). Example 72: If n has a primitive root, then the number of quadratic residues of n is φ(n) 2. Example 73: If n has a primitive root r, then the quadratic residues of n are exactly the even powers of r. Example 74: If n has a primitive root r, then the product of two non-quadratic residues is a quadratic residue. November 13 th. The quadratic reciprocity. Example 75: Theorem Example 76: Theorem Example 77: Exercise 10.20(a). Example 78: Exercise Example 79: Exercise November 15 th. The quadratic reciprocity. Worksheet on Chapter 10. Example 80: Proof of Corollary Example 81: Proof of Theorem November 20 th. Multiplicative functions. σ and τ. Example 82: Example 11.3 and Exercise 11.5(e). Example 83: Theorem (one way only). Example 84: Theorem Example 85: Lemma Example 86: Exercise 11.28(a) (incomplete). 4

5 November 27 th. The Möbius inversion formula. Example 87: Exercise 11.28(a) (the other half). Example 88: Exercise Example 89: Exercise November 29 th. Fermat primes and Mersenne primes. Worksheet on Chapter 11. Example 90: Lemma Example 91: Theorem December 4 th. Review and missing proofs. Example 92: Theorem Example 93: Lemma December 6 th. Review.. Exam 4 (Chapters 10, 11, and 12). December 7 th and 8 th. Consultation Days. Office hours: Thursday: 8:00-10:00 AM and 3:00-5:00 PM. Friday: 2:00-4:00 PM. December 13 th. Final Exam: 08:45-10:45AM in S 139. This exam is comprehensive, so it tests Chapters 2-12, except Sections 3.4 and