By Marlow Anderson

A part of the PWS complicated arithmetic sequence, this article comprises chapters on polynomials and factoring, specific factorization, ring homomorphisms and beliefs, and constructibility difficulties and box extensions.

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The proof that multiplication on Zm is also well defined is similar and is left as Exercise 9. ✷ We now have an ‘arithmetic’ defined on Zm . To avoid cumbersome notation, it is common to write the elements of Zm as simply 0, 1, . . , m − 1 instead of [0], [1], . . , [m − 1]. So, in Z5 , 3 + 4 = 2 and 2 + 3 = 0. ) Bear in mind that the arithmetic is really on residue classes. For the remainder of this chapter we will not omit the brackets, although later we often will. 4 A first simple example of this arithmetic is in the case where m = 2.

Suppose that a and b are positive integers. If a + b is prime, prove that gcd(a, b) = 1. 22 Exercises 7. (a) A natural number greater than 1 that is not prime is called composite. Show that for any n, there is a run of n consecutive composite numbers. Hint: Think factorial. (b) Therefore, there is a string of 5 consecutive composite numbers starting where? 8. Prove that two consecutive members of the Fibonacci sequence are relatively prime. 9. Notice that gcd(30, 50) = 5 gcd(6, 10) = 5 · 2. In fact, this is always true; prove that if a = 0, then gcd(ab, ac) = a · gcd(b, c).

But this temporary perversity now will allow us to be consistent with the more general terminology we’ll use later. We reserve the term ‘prime’ for another definition: An integer p (other than 0 and ±1) is prime if, whenever p divides ab, then either p divides a or p divides b. (Notice that when we say ‘or’ here, we mean one or the other or both. 6 For instance, we know that 2 is a prime integer. For if 2|ab, then ab is even. But a product of integers is even exactly if at least one of the factors is even, and so 2|a or 2|b.