Week 9 Exercises, due 23 November 2021

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Exercises 8.9 Week 9 Exercises, due 23 November 2021

Define the Fibonacci numbers \(f_n\) for \(n\in\NN_0\) recursively as follows. We set \(f_0=0\) and \(f_1 = 1\text{.}\) Then, for every \(n\in\NN_0\) with \(n \geq 2\text{,}\) we set \(f_n = f_{n-1}+f_{n-2}\text{.}\) We are going to establish some divisibility properties of these numbers, using the Euclidean algorithm and the Bezout theorem.

1.

Prove that for every triple of integers \(a,b,c\in\ZZ\text{,}\) one has \(\gcd(a,b)=\gcd(a,ac+b)\text{.}\)

2.

Show that for every \(n\in\NN_0\text{,}\) the Fibonacci numbers \(f_{n}\) and \(f_{n+1}\) are coprime.

3.

Prove that for every \(r,s\in\NN\text{,}\) one has \(f_{r+s} = f_rf_{s+1} + f_{r-1}f_s\text{.}\)

4.

Prove that for every \(r,s\in \NN_0\text{,}\) the Fibonacci number \(f_r\) divides the Fibonacci number \(f_{rs}\text{.}\)

5.

Prove that for every triple of integers \(a,b,c\in\ZZ\text{,}\) if \(a\) and \(c\) are coprime, then \(\gcd(a,b)=\gcd(a,bc)\text{.}\)

6.

Show that for any \(r,s\in \NN_0\text{,}\) one has \(\gcd(f_r,f_s)=f_{\gcd(r,s)}\text{.}\)

7.

For which pairs \(r,s\in \NN_0\) does \(f_r\) divide \(f_s\text{?}\)