# A COMPUTATIONAL INTRODUCTION TO NUMBER THEORY AND ALGEBRA by VICTOR SHOUP

By VICTOR SHOUP

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Extra info for A COMPUTATIONAL INTRODUCTION TO NUMBER THEORY AND ALGEBRA (VERSION 1)

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13. Suppose that f and g are piece-wise continuous [a, ∞), both x of which are eventually positive. For x ≥ a, deﬁne F (x) := a f (t)dt and x G(x) := a g(t)dt. Show that if f ∼ g and G(x) → ∞ as x → ∞, then F ∼ G. 2 Machine models and complexity theory When presenting an algorithm, we shall always use a high-level, and somewhat informal, notation. However, all of our high-level descriptions can be routinely translated into the machine-language of an actual computer. So that our theorems on the running times of algorithms have a precise mathematical meaning, we formally deﬁne an “idealized” computer: the random access machine or RAM.

Suppose that f and g are functions deﬁned on the integers k, k + 1, . , both of which are eventually positive. For n ≥ k, deﬁne F (n) := n n i=k f (i) and G(n) := i=k g(i). Show that if f ∼ g and G(n) → ∞ as n → ∞, then F ∼ G. The following two exercises are continuous variants of the previous two exercises. ” In particular, we restrict ourselves to piecewise continuous functions (see §A3). 12. Suppose that f and g are piece-wise continuous on [a, ∞), x and that g is eventually positive. For x ≥ a, deﬁne F (x) := a f (t)dt and x G(x) := a g(t)dt.

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 . 1, in every iteration of the loop in lines 13–15, the value of carry is 0 or 1, and the value tmp lies between 0 and 2B − 1.