By VICTOR SHOUP
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Extra resources for A COMPUTATIONAL INTRODUCTION TO NUMBER THEORY AND ALGEBRA (VERSION 1)
Now we turn to the general case ≥ 1. 2. Let x and y be integers such that 0 ≤ x = x 2n + s and 0 < y = y 2n + t for some integers n, s, t, x , y with n ≥ 0, 0 ≤ s < 2n , and 0 ≤ t < 2n . Further suppose that 2y ≥ x/y. Then we have x/y ≤ x /y ≤ x/y + 2. Proof. For the ﬁrst inequality, note that x/y ≤ x/(y 2n ), and so x/y ≤ x/(y 2n ) , and by the previous theorem, x/(y 2n ) = x /y . That proves the ﬁrst inequality. For the second inequality, ﬁrst note that from the deﬁnitions, x/y ≥ x /(y +1), which is equivalent to x y−xy −x ≤ 0.
12 simply says that φ is multiplicative. In this section, we develop some of the theory of arithmetic functions that is pertinent to number theory; however, the results in this section will play only a very minor role in the remainder of the text. We begin with a simple observation, which the reader may easily verify: if f is a multiplicative function, and if n = pe11 · · · perr is the prime factorization of n, then f (n) = f (pe11 ) · · · f (perr ). Next, we deﬁne a binary operation on arithmetic functions that has a number of interesting properties and applications.
13) if and only if 2 d|n (µ(d)) φ(d) = n. 27. Show that for any arithmetic function f with f (1) = 0, there is a unique arithmetic function g, called the Dirichlet inverse of f , such that f g = I. Also, show that if f (1) = 0, then f has no Dirichlet inverse. 28. Show that if f is a multiplicative function, then so is its Dirichlet inverse (as deﬁned in the previous exercise). TEAM LinG 3 Computing with large integers In this chapter, we review standard asymptotic notation, introduce the formal computational model we shall use throughout the rest of the text, and discuss basic algorithms for computing with large integers.
A COMPUTATIONAL INTRODUCTION TO NUMBER THEORY AND ALGEBRA (VERSION 1) by VICTOR SHOUP