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Question - Math 324 Assignment 2 (due September 24)

1. (1.5, #44)(5) Show that the nth Fibonacci number fn, is even if, and only if, 3 divides n.

2. (3.3, #6)(5) If (a, b) =1 and (a, c) = 1, show that (a, bc) = 1.

3. (2 + 3 + 2) With a = 2639 and b = 2145, find
i) (a, b), ii) x, y with ax + by = (a, b), and iii) [a, b].

4. (3.5, #8)(5) Show that every positive integer is the product of a square integer and a
squarefree integer. (An integer is squarefree if it is not divisible by any square > 1).

5. (6)(Continuing problem 1.5) Is the factorization of every L-number into L-primes
unique? Why?

6. (6) Show that | ∑ 1 ≤ n ≤ N 1/n − log(N) | is a bounded function o ...Read More f N ≥ 1. Hint: Apply the
method at the beginning of the proof of (D1.1).

7. (2 + 2 + 2) Consider the set A of all numbers of the form α = a + b6−with a, b in Z.
i) Show that A is a subring of the complex number field C.
ii) Define the norm N(α) = α⋅α with αthe complex conjugate of α∈ A. Show that
N(αβ) = N(α)N(β) for all α, β in A, and that N(a + b6−) = a2 + 6b2.
iii) Show that α ∈ A is a unit of A if, and only if, N(α) = 1. ...Read Less

Solution Preview - f of the form 3k, 3k + 1, or 3k +2 it suffices to show that f 3k, f 3k +1 − 1, and f 3k + 2 − 1 are all even for every integer k ≥ 0. Proof by induction on k with base case true since f0, f1 − 1, f2 − 1 are all 0. Let k ≥ 1, and assume the induction hypothesis for k − 1. Then (i) f3k = f3k − 1 + f3k − 2 = (f3(k − 1) +2 −

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