CS代写 CSI 2101: Discrete Structures

Number Theory (Part B)
CSI 2101: Discrete Structures
School of Electrical Engineering and Computer Science, University of Ottawa
February 03, 2022

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Md. Hasan (uOttawa) Discrete Structures 4b MdH W22 February 03, 2022 1 / 16

1 Properties of Primes
2 Greatest Common Divisors
3 Linear Congruences
Md. Hasan (uOttawa) Discrete Structures 4b MdH W22 February 03, 2022 2 / 16

Trial Division and The Sieve of Eratosthenes
􏰀 If n is a composite integer, then n has a prime divisor less than or equal √
􏰀 An integer is prime if it is not divisible by any prime less than or equal to its square root. This leads to the brute-force algorithm known as trial division.
􏰀 The sieve of Erastosthenes is used to find all primes not exceeding a specified positive integer. It starts with a list of integers. The sieving process continues for all primes ≤ √n where n is a given positive integer.
Md. Hasan (uOttawa) Discrete Structures 4b MdH W22 February 03, 2022 3 / 16

􏰀 The Infinitude of Primes There are infinitely many primes.
􏰀 The Prime Number Theorem
The ratio of π(x), the number of primes not exceeding x, and x/ln(x)
approaches 1 as x grows without bound. 􏰀 Mersene Primes
Prime numbers of the form 2p − 1, where p is prime, are called Mersene primes.
Md. Hasan (uOttawa) Discrete Structures 4b MdH W22 February 03, 2022 4 / 16

Conjectures About Primes
􏰀 Goldbach’s Conjecture
Every even integer n, n > 2, is the sum of two primes.
􏰀 Primes of the form n2 + 1
There are infinitely many primes of the form n2 + 1, where n is a positive integer. It has been shown that there are infinitely many primes of the form n2 + 1, where n is a positive integer or the product of at most two primes.
􏰀 The Twin Prime Conjecture
The twin prime conjecture asserts that there are infinitely many pairs of
twin primes. Twin primes are pairs of primes that differ by 2.
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Greatest Common Divisors
􏰀 Definition
Let a and b be integers, not both zero. The largest integer d such that d | a and d | b is called the greatest common divisor of a and b. the greatest common divisor of a and b is denoted by gcd(a, b).
• gcd(27, 45) = 9 • gcd(11, 17) = 1
􏰀 Relatively Prime
The integers a and b are relatively prime if their greatest common divisor
Md. Hasan (uOttawa) Discrete Structures 4b MdH W22 February 03, 2022 6 / 16

Greatest Common Divisors (cont.)
􏰀 Pairwise Relatively Prime
The integers a,a,…,an are pairwise relatively prime if gcd(ai,aj) = 1
whenever 1 ≤ i < j ≤ n. – The integers 10, 17, and 21 are pairwise relatively prime. – The integers 10, 19, and 24 are not pairwise relatively prime. 􏰀 Prime Factorization and GCD a = pa pa ...pan b = pb pb ...pbn n gcd(a,b) = pmin(a,b)pmin(a,b)...pmin(an,bn) n Md. Hasan (uOttawa) Discrete Structures 4b MdH W22 February 03, 2022 7 / 16 Greatest Common Divisors (cont.) 􏰀 Least Common Multiple The least common multiple of the positive integers a and b is the smallest positive integer that is divisible by both a and b. The least common multiple of a and b is denoted by lcm(a, b). 􏰀 lcm(a, b) = pmax(a,b)pmax(a,b)...pmax(an,bn) n 􏰀 Let a and b be positive integers. Then ab =gcd(a, b)·lcm(a, b). Md. Hasan (uOttawa) Discrete Structures 4b MdH W22 February 03, 2022 8 / 16 The Euclidean Algorithm 􏰀 The prime factorization can be a computationally inefficient method of finding the GCDs. It takes time to factorize primes. 􏰀 The Euclidean algorithm is a more efficient method of finding the GCDs. 􏰀 =bq+r,wherea,b,q,andr areintegers. Then gcd(a, b)=gcd(b, r ). If an integer d divides both a and b, then it also divides a − bq = r . Thus, any common divisor of a and b is also a common divisor of b and r. Likewise, if an integer d divides both b and r, then it also divides bq + r = a. Thus, any common divisor of b and r is also a common divisor of a and b. Consequently, gcd(a,b)=gcd(b,r). Md. Hasan (uOttawa) Discrete Structures 4b MdH W22 February 03, 2022 9 / 16 The Euclidean Algorithm (cont.) 􏰀 The Euclidean Algorithm (a ≥ b) procedure gcd (a, b : positive integers) x := a while y ̸= 0 r := x mod y x := y return x{gcd(a,b) is x} Md. Hasan (uOttawa) Discrete Structures 4b MdH W22 February 03, 2022 GCDs as Linear Combinations 􏰀 Bezout’s Theorem/Identity If a and b are positive integers, then there exist integers s and t such that gcd(a, b) = sa + tb. Here, s and t are called Bezout’s coefficients of a and b. Express gcd(252, 198) = 18 as a linear combination of 252 and 198. Md. Hasan (uOttawa) Discrete Structures 4b MdH W22 February 03, 2022 11 / 16 GCDs as Linear Combinations (cont.) 􏰀 Method 1: Backward Euclidean Suppose that a and b are positive integers with a ≥ b. Let r0 = a and r0 = r1q1 + r2 0 ≤ r2 < r1, r1 = r2q2 + r3 0 ≤ r3 < r2, . rn−2 =rn−1qn−1 +rn rn−1 = rnqn. 0≤rn 1, then an inverse of a modulo m exists. In addition, if a ̄ where a ̄ < m, is an inverse of a modulo m, then every other inverse of a modulo m is congruent to a ̄ modulo m. Md. Hasan (uOttawa) Discrete Structures 4b MdH W22 February 03, 2022 15 / 16 Thank You! Questions and Comments? Md. Hasan (uOttawa) Discrete Structures 4b MdH W22 February 03, 2022 16 / 16 程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com