Abstract Algebra Theory and Applications - Computer Science ...

1.2 THE DIVISION ALGORITHM 29To find r and s such that ar + bs = d, we begin with this last equation andsubstitute results obtained from the previous equations:d = r n= r n−2 − r n−1 q n= r n−2 − q n (r n−3 − q n−1 r n−2 )= −q n r n−3 + (1 + q n q n−1 )r n−2.= ra + sb.The algorithm that we have just used to find the greatest common divisord of two integers a and b and to write d as the linear combination of a andb is known as the Euclidean algorithm.Prime NumbersLet p be an integer such that p > 1. We say that p is a prime number, orsimply p is prime, if the only positive numbers that divide p are 1 and pitself. An integer n > 1 that is not prime is said to be composite.Lemma 1.6 (Euclid) Let a and b be integers and p be a prime number. Ifp | ab, then either p | a or p | b.Proof. Suppose that p does not divide a. We must show that p | b. Sincegcd(a, p) = 1, there exist integers r and s such that ar + ps = 1. Sob = b(ar + ps) = (ab)r + p(bs).Since p divides both ab and itself, p must divide b = (ab)r + p(bs).□Theorem 1.7 (Euclid) There exist an infinite number of primes.Proof. We will prove this theorem by contradiction. Suppose that thereare only a finite number of primes, say p 1 , p 2 , . . . , p n . Let p = p 1 p 2 · · · p n +1.We will show that p must be a different prime number, which contradictsthe assumption that we have only n primes. If p is not prime, then it mustbe divisible by some p i for 1 ≤ i ≤ n. In this case p i must divide p 1 p 2 · · · p nand also divide 1. This is a contradiction, since p > 1.□