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3-2. 1 are multiplicative. 3-3. For each r ∈ N0 define the arithmetic function [r] : Z+ −→ R by [r](n) = nr . In particular, [0] = η and [1] = id. a) Show that [r] is multiplicative. b) If r > 0, show that σr = [r] ∗ η. Deduce that σr is multiplicative. c) Show that [r] ∗ [r] satisfies [r] ∗ [r](n) = nr τ (n). d) Find a general formula for [r] ∗ [s](n) when s < r. 3-4. For n ∈ Z+ , prove the following formulæ, where the functions are defined in the text or in earlier questions. µ(d)σ(n/d) = n; (a) d|n µ(d)τ (n/d) = 1; (b) d|n σr (d)µ(n/d) = nr .

A) Prove the identities n+ n2 + 1 = 2n + ( n2 + 1 − n) = 2n + 1 √ . n + n2 + 1 √ √ b) Show that [ n2 + 1] = n and that the infinite continued fraction expansion of n2 + 1 is [n; 2n]. √ √ c) Show that [ n2 + 2] = n and that the infinite continued fraction expansion of n2 + 2 is [n; n, 2n]. 28 1. BASIC NUMBER THEORY √ d) Show that [ n2 + 2n] = n and that the infinite continued fraction expansion of √ n2 + 2n is [n; 1, 2n]. 1-23. Find the fundamental solutions of Pell’s equation x2 − dy 2 = 1 for each of the values d = 5, 6, 8, 11, 12, 13, 31, 83.

4-2. Show that a subset of a countable set is countable. 4-3. Let X be a countable set. If Y is a finite set, show that the cartesian product X × Y = {(x, y) : x ∈ X, y ∈ Y } is countable. 10(d) or a modification of its proof to show that this is still true if Y is countably infinite.

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Algebra in a Local Topos With Applns to Ring Theory by F. Borceux, et al.,


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