Number Theory for Computing by Song Y. Yan

By Song Y. Yan

This ebook presents a great creation to the classical simple quantity thought and the fashionable algorithmic quantity conception, and their functions in computing and data expertise, together with desktops layout, cryptography and community defense. during this moment variation proofs of many theorems were supplied, extra additions and corrections have been made.

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1 (The Sieve of Eratosthenes). Given a positive integer n > I, this algorithm will find all prime numbers up to n. [1] Create a list of integers from 2 to n; [2] For prime numbers Pi (i = 1,2, ... ) from 2,3,5 up to multiples Pi < Pim :::; n from the list; [3] Print the integers remaining in the list. l v'nJ, delete all the 1. 2 FundaIIlental TheoreIIl of ArithIIletic First, let us investigate a simple but important property of composite numbers. 5. Every composite number has a prime factor. Proof.

Qn-I, qn]. 60) If each qi is an integer, the continued fraction is called simple; a simple continued fraction can either be finite or infinite. A continued fraction formed from [qo, ql, q2, ... qn-I ,qn] by neglecting all of the terms after a given term is called a convergent of the original continued fraction. If we denote the k-th Pk convergent by C k = Qk' then C - Po _ qo. 0- Qo - (1) l' Cl = PI = qOql + 1 Ql ql + Pk- 2 ,lor c k 2 > . + Qk-2 (2) If Pk = qkQk-1 + Qk-2 and Qk = qkPk-l + Pk- 2, then gcd(Pk, Qk) = 1.

4. Let a and b be integers, not both zero. The largest divisor d such that dia and d Ibis called the greatest common divisor (gcd) of a and b. The greatest common divisor of a and b is denoted by gcd(a, b). 5. The sets of positive divisors of 111 and 333 are as follows: 1,3,37,111, 1,3,9,37,111,333, so gcd(111,333) = 111. But gcd(91,111) = 1, since 91 and 111 have no common divisors other than 1. 5. Integers a and bare called relatively prime if gcd(a, b) = 1. We say that integers nl, n2, ... , nk are pairwise relatively prime if, whenever i =I j, we have gcd(ni,nj) = 1.

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