Dietzfelbinger M. Primality testing in polynomial time ... - tiera.ru

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44 3. Fundamentals from Number Theory Proof. Choose prime decompositions p1 ···pr of n and q1 ···qs of m. Then p1 ···pr · q1 ···qs is a prime decomposition of n · m. By Lemma 3.5.7 and the fundamental theorem (Theorem 3.5.8) p appears among p1,...,pr,q1,...,qr. If p is one of the pi, thenp divides n, otherwise it is among the qj’s and hence divides m. ⊓⊔ Corollary 3.5.10. If p1,...,pr are distinct prime numbers that all divide n, then p1 ···pr divides n. Proof. Let n = q1 ···qs be the prime decomposition of n. Foreachpi we have the following: By Lemma 3.5.7 and Theorem 3.5.8, pi must occur among q1,...,qs. Nowthepi are distinct, so by reordering we may assume pi = qi, for 1 ≤ i ≤ r. Thenn =(p1 ···pr) · (qr+1 ···qs), which proves the claim. ⊓⊔ We close this section with a remark on the connection between the concept of numbers being relatively prime and their prime factorization and draw a consequence for the problem of calculating ϕ(n). Proposition 3.5.11. Let n, m ≥ 1 have prime decompositions n = p1 ···pr and m = q1 ···qs. Thena, b are relatively prime if and only if {p1,...,pr}∩ {q1,...,qs} = ∅. Proof. “⇒”: Indirectly. Suppose a prime number p occurred in both prime decompositions. Then p would divide both n and m, hence gcd(n, m) would be larger than 1. — “⇐”: Indirectly. Suppose gcd(n, m) > 1. Then there is a prime number p that divides gcd(n, m) and hence divides both n and m. By Theorem 3.5.8 p occurs in {p1,...,pr} and in {q1,...,qs}. ⊓⊔ Corollary 3.4.5 stated that ϕ(n1 · n2) =ϕ(n1) · ϕ(n2) ifn1 and n2 are relatively prime. This makes it possible to calculate ϕ(n) forn easily once the prime decomposition of n is known. Proposition 3.5.12. If n ≥ 1 has the prime decomposition n = p k1 1 ···pkr r for distinct prime numbers p1,...,pr, then ϕ(n) = � (pi − 1)p ki−1 = n · � � 1 − 1 � . 1≤i≤r i 1≤i≤r Proof. The formulas are trivially correct for n = 1. Thus, assume n ≥ 2. Case 1: n is a prime number. — Then ϕ(n) =|{1,...,n− 1}| = n − 1, and the formulas are correct. Case 2: n = p k for a prime number p and some k ≥ 2. — Then ϕ(n) =|{a | 1 ≤ a
ϕ(n) = � 3.6 Chebychev’s Theorem on the Density of Prime Numbers 45 1≤i≤r ϕ(p ki i ), which by Case 2 means ϕ(n) = � (pi − 1) · p 1≤i≤r ki−1 = � p 1≤i≤r ki � · 1 − 1 � = n · pi � � 1 − 1≤i≤r 1 � . pi ⊓⊔ For example, we have ϕ(7) = 6; ϕ(15) = ϕ(3 · 5) = 2 · 4=8=15· 2 4 3 · 5 , ϕ(210) = ϕ(2 · 3 · 5 · 7) = 1 · 2 · 4 · 6 = 48 = 210 · 1 2 4 6 2 · 3 · 5 · 7 , ϕ(1000) = ϕ(23 · 53 )=1· 22 · 4 · 52 = 400 = 1000 · 1 4 2 · 5 , ϕ(6860) = ϕ(22 · 5 · 73 )=1· 2 · 4 · 6 · 72 = 2352 = 6860 · 1 4 6 2 · 5 · 7 . 3.6 Chebychev’s Theorem on the Density of Prime Numbers In this section, we develop upper and lower bounds on the density of the prime numbers in the natural numbers. Up to now, we only know that there are infinitely many prime numbers. We want to estimate how many primes there are up to some bound x. Results of the type given here were first proved by Chebychev in 1852, and so they are known as “Chebychev-type estimates”. Definition 3.6.1. For x>1, letπ(x) denote the number of primes p ≤ x. Table 3.5 lists some values of this function, and compares it with the function x/(ln x − 1). x 2 3 4 5 6 7 8 9 10 11 12 13 14 15 20 30 π(x) 1 2 2 3 3 4 4 4 4 5 5 6 6 6 8 10 x 40 50 70 100 200 500 1000 5000 10000 π(x) 12 15 19 25 46 95 168 669 1229 x/(ln x − 1) 14.9 17.2 21.5 27.7 46.5 95.9 169.3 665.1 1218.0 Table 3.5. The prime counting function π(x) for some integral values of x; values of the function x/(ln x − 1) (rounded) in comparison The following theorem is important, deep, and famous; it was conjectured as early as in the 18th century by Gauss, but proved only in 1896, independently by Hadamard and de la Vallée Poussin.
Page 1 and 2: Lecture Notes in Computer Science 3
Page 3 and 4: Martin Dietzfelbinger Primality Tes
Page 5 and 6: To Angelika, Lisa, Matthias, and Jo
Page 7 and 8: VIII Preface toundingly it gets by
Page 9 and 10: X Contents 5. The Miller-Rabin Test
Page 11 and 12: 2 1. Introduction: Efficient Primal
Page 19 and 20: 10 1. Introduction: Efficient Prima
Page 21 and 22: 12 1. Introduction: Efficient Prima
Page 23 and 24: 14 2. Algorithms for Numbers and Th
Page 31 and 32: 3. Fundamentals from Number Theory
Page 33 and 34: 3.1 Divisibility and Greatest Commo
Page 35 and 36: 3.2 The Euclidean Algorithm 27 Prop
Page 37 and 38: 3.2 The Euclidean Algorithm 29 (b)
Page 39 and 40: 3.2 The Euclidean Algorithm 31 We n
Page 41 and 42: 3.3 Modular Arithmetic 33 Lemma 3.3
Page 43 and 44: 3.4 The Chinese Remainder Theorem 3
Page 45 and 46: 3.4 The Chinese Remainder Theorem 3
Page 47 and 48: 3.5 Prime Numbers 39 3.5.1 Basic Ob
Page 49 and 50: 3.5 Prime Numbers 41 steps in the v
Page 51: 3.5 Prime Numbers 43 r ≥ 0. Clear
Page 55 and 56: 3.6 Chebychev’s Theorem on the De
Page 63 and 64: 56 4. Basics from Algebra: Groups,
Page 79 and 80: 5. The Miller-Rabin Test In this ch
Page 81 and 82: 5.1 The Fermat Test 75 multiples of
Page 83 and 84: 5.1 The Fermat Test 77 the set {n |
Page 85 and 86: 5.2 Nontrivial Square Roots of 1 79
Page 87 and 88: 5.2 Nontrivial Square Roots of 1 81
Page 89 and 90: Lemma 5.3.1. (a) L A n ⊆ BA n . (
Page 91 and 92: 6. The Solovay-Strassen Test The pr
Page 93 and 94: 6.2 The Jacobi Symbol 87 Definition
Page 95 and 96: a 6.3 The Law of Quadratic Reciproc
Page 97 and 98: 6.3 The Law of Quadratic Reciprocit
Page 99 and 100: 6.4 Primality Testing by Quadratic
Page 101 and 102: 7. More Algebra: Polynomials and Fi
Page 103 and 104:
7.1 Polynomials over Rings 97 Defin
Page 105 and 106:
7.1 Polynomials over Rings 99 Remar
Page 107 and 108:
7.1 Polynomials over Rings 101 (b)
Page 109 and 110:
7.2 Division with Remainder and Div
Page 111 and 112:
7.3 Quotients of Rings of Polynomia
Page 113 and 114:
7.3 Quotients of Rings of Polynomia
Page 115 and 116:
7.4 Irreducible Polynomials and Fac
Page 117 and 118:
7.5 Roots of Polynomials 111 X, has
Page 119 and 120:
7.6 Roots of the Polynomial X r −
Page 121 and 122:
8. Deterministic Primality Testing
Page 123 and 124:
8.2 The Algorithm of Agrawal, Kayal
Page 125 and 126:
8.3 The Running Time 119 Time for t
Page 127 and 128:
8.3 The Running Time 121 Lemma 8.3.
Page 129 and 130:
8.5 Proof of the Main Theorem 123 B
Page 131 and 132:
8.5 Proof of the Main Theorem 125 L
Page 133 and 134:
8.5 Proof of the Main Theorem 127 P
Page 135 and 136:
8.5 Proof of the Main Theorem 129 (
Page 137 and 138:
8.5 Proof of the Main Theorem 131 i
Page 139 and 140:
134 A. Appendix Proof. For k < 0ork
Page 141 and 142:
136 A. Appendix as an abbreviation
Page 143 and 144:
138 A. Appendix a 1 2 3 4 5 6 7 8 9
Page 145 and 146:
140 A. Appendix Lemma A.3.3. If p
Page 147 and 148:
142 A. Appendix Induction step: Ass
Page 149 and 150:
144 References 20. Gauss, C.F., Dis
Page 151 and 152:
146 Index efficient algorithm, 2 eq
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Dietzfelbinger M. Primality testing in polynomial time ... - tiera.ru

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