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By VICTOR SHOUP

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2. In line 5, we reduce qi if it is obviously too big. 3. In lines 6–10, we compute (ri+ · · · ri )B ← (ri+ · · · ri )B − qi b. In each loop iteration, the value of tmp lies between −(B 2 − B) and B − 1, and the value carry lies between −(B − 1) and 0. 4. If the estimate qi is too large, this is manifested by a negative value of ri+ at line 10. Lines 11–17 detect and correct this condition: the loop body here executes at most twice; in lines 12–16, we compute (ri+ · · · ri )B ← (ri+ · · · ri )B + (b −1 · · · b0 )B .

5. Suppose that x takes non-negative integer values, and that g(x) > 0 for all x ≥ x0 for some x0 . Show that f = O(g) if and only if |f (x)| ≤ cg(x) for some positive constant c and all x ≥ x0 . 6. Give an example of two non-decreasing functions f and g, both mapping positive integers to positive integers, such that f = O(g) and g = O(f ). 7. Show that (a) the relation “∼” is an equivalence relation on the set of eventually positive functions; (b) for eventually positive functions f1 , f2 , g2 , g2 , if f1 ∼ f2 and g1 ∼ g2 , then f1 g1 ∼ f2 g2 , where “ ” denotes addition, multiplication, or division; (c) for eventually positive functions f1 , f2 , and any function g that tends to infinity as x → ∞, if f1 ∼ f2 , then f1 ◦ g ∼ f2 ◦ g, where “◦” denotes function composition.

An algebraic structure satisfying the conditions in the above theorem is known more generally as a “commutative ring with unity,” a notion that we will discuss in Chapter 9. Note that while all elements of Zn have an additive inverses, not all elements of Zn have a multiplicative inverse. 4, holds if and only if gcd(a, n) = 1. 5), it follows that if α ∈ Zn has a multiplicative inverse in Zn , then this inverse is unique, and we may denote it by α−1 . One denotes by Z∗n the set of all residue classes that have a multiplicative inverse.

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A COMPUTATIONAL INTRODUCTION TO NUMBER THEORY AND ALGEBRA (VERSION 1) by VICTOR SHOUP


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