By VICTOR SHOUP
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Extra info for A COMPUTATIONAL INTRODUCTION TO NUMBER THEORY AND ALGEBRA (VERSION 1)
2) are precisely those integers that are congruent to 3 modulo 7, which we can list as follows: . . , −18, −11, −4, 3, 10, 17, 24, . . ✷ In the next section, we shall give a systematic treatment of the problem of solving linear congruences, such as the one appearing in the previous example. 1. Let x, y, n ∈ Z with n > 0 and x ≡ y (mod n). Also, let a0 , a1 , . . , ak be integers. Show that a0 + a1 x + · · · + ak xk ≡ a0 + a1 y + · · · + ak y k (mod n). 2. Let a, b, n, n ∈ Z with n > 0 and n | n.
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. Show that if f = O(g) and G is eventually positive, then F = O(G).
We could have inferred this fact from the theorem, since gcd(2, 15) = 1. In the third row, the only numbers hit are the multiples of 3, which follows from the theorem and the fact that gcd(3, 15) = 3. Also note that the pattern in this row repeats every ﬁve columns; that is also implied by the theorem; that is, 3z ≡ 3z (mod 15) if and only if z ≡ z (mod 5). In the fourth row, we again see a permutation of the ﬁrst row, which follows from the theorem and the fact that gcd(4, 15) = 1. In the ﬁfth row, the only numbers hit are the multiples of 5, which follows from the theorem and the fact that gcd(5, 15) = 5.
A COMPUTATIONAL INTRODUCTION TO NUMBER THEORY AND ALGEBRA (VERSION 1) by VICTOR SHOUP