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Unformatted text preview: 10 Computational Problems Related to Quadratic Residues 61 10.1 Computing the Jacobi Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 10.2 Testing quadratic residuosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 10.3 Computing modular square roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 11 Primality Testing 65 11.1 Trial Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 11.2 A Fast Probabilistic Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 11.3 The Distribution of Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 11.4 Deterministic Primality Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 iv Chapter 1 Basic Properties of the Integers This chapter reviews some of the basic properties of the integers, including notions of divisibility and primality, unique factorization into primes, greatest common divisors, and least common multiples. 1.1 Divisibility and Primality Consider the integers Z = { ..., 1 , , 1 , 2 ,... } . For a,b ∈ Z , we say that b divides a , and write b  a , if there exists c ∈ Z such that a = bc . If b  a , then b is called a divisor of a . If b does not divide a , then we write b a . We first state some simple facts: Theorem 1.1 For all a,b,c ∈ Z , we have 1. a  a , 1  a , and a  ; 2.  a if and only if a = 0 ; 3. a  b and b  c implies a  c ; 4. a  b implies a  bc ; 5. a  b and a  c implies a  b + c ; 6. a  b and b  a if and only if a = ± b . Proof. Exercise. 2 We say that an integer p is prime if p > 1 and the only divisors of p are ± 1 and ± p . Conversely, and integer n is called composite if n > 1 and it is not prime. So an integer n > 1 is composite if and only if n = ab for some integers a,b with 1 < a,b < n . A fundamental fact is that any integer can be written as a signed product of primes in an essentially unique way. More precisely: Theorem 1.2 Every nonzero integer n can be expressed as n = ± Y p p ν p ( n ) , where the product is over all primes, and all but a finite number of the exponents are zero. Moreover, the exponents and sign are uniquely determined by n . 1 To do prove this theorem, we may clearly assume that n is positive. The proof of the existence part of Theorem 1.2 is easy. If n is 1 or prime, we are done; otherwise, there exist a,b ∈ Z with 1 < a,b < n and n = ab , and we apply an inductive argument with a and b . The proof of the uniqueness part of Theorem 1.2 is not so simple, and most of the rest of this chapter is devoted to developing the ideas behind such a proof. The essential ingredient in the proof is the following: Theorem 1.3 (Division with Remainder Property) For a,b ∈ Z with b > , there exist unique q,r ∈ Z such that a = bq + r and ≤ r < b ....
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 Spring '13
 MRR
 Math, Algebra, Number Theory

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