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Let n be a positive integer. 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.

This follows from the definition and the fact that the prime power factorizations of two coprime natural numbers m, n have no common prime factors. So for example, µ(105) = µ(3)µ(5)µ(7) = (−1)3 = −1. 47 48 3. 3. The M¨ obius function µ satisfies µ(1) = 1, µ(d) = 0 if n 2. d|n Proof. By Induction on r, the number of prime factors in the prime power factorization of n = pr11 · · · prt t , so r = r1 + · · · + rt . If r = 1, then n = p is prime and µ(p) = −1, hence µ(d) = 1 − 1 = 0. d|p Assume that whenever r < k.

A) Show that ϕE is an edge and that this defines an action of G on X. b) Find OrbG (E) and StabG (E) for the edge AB. c) For each of the following elements of G find FixG (g): ι, (A B), (A B C), (A B C D), (A B)(C D). 2-13. Let X = {1, 2, 3, 4, 5, 6, 7, 8} and G = (1 2 3 4 5 6)(7 8) be the cyclic subgroup of S7 acting on X in the obvious way. How many orbits does this action of G have? 2-14. How many distinguishable 5-bead circular necklaces can be made where each bead has to be a different colour chosen from 5 colours?