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# 数学代写|离散数学代写DISCRETE MATHEMATICS代写|MATH200 Fibonacci and Lucas Numbers; Mersenne Primes

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## 数学代写|离散数学代写DISCRETE MATHEMATICS代写|Fibonacci and Lucas Numbers; Mersenne Primes

We have encountered the Fibonacci numbers (after Leonardo Fibonacci, also known as Leonardo of Pisa, 1170-1250) in Section 2.3. These numbers show up unexpectedly in many places, including algorithm design and analysis, for example, Fibonacci heaps. The Lucas numbers (after Edouard Lucas, 1842-1891) are closely related to the Fibonacci numbers. Both arise as special instances of the recurrence relation
$$u_{n+2}=u_{n+1}+u_{n}, n \geq 0$$
where $u_{0}$ and $u_{1}$ are some given initial values.

The Fibonacci sequence $\left(F_{n}\right)$ arises for $u_{0}=0$ and $u_{1}=1$, and the Lucas sequence $\left(L_{n}\right)$ for $u_{0}=2$ and $u_{1}=1$. These two sequences turn out to be intimately related and they satisfy many remarkable identities. The Lucas numbers play a role in testing for primality of certain kinds of numbers of the form $2^{p}-1$, where $p$ is a prime, known as Mersenne numbers. In turns out that the largest known primes so far are Mersenne numbers and large primes play an important role in cryptography.
It is possible to derive a closed-form formula for both $F_{n}$ and $L_{n}$ using some simple linear algebra.
Observe that the recurrence relation
$$u_{n+2}=u_{n+1}+u_{n}$$
yields the recurrence
$$\left(\begin{array}{c} u_{n+1} \ u_{n} \end{array}\right)=\left(\begin{array}{ll} 1 & 1 \ 1 & 0 \end{array}\right)\left(\begin{array}{c} u_{n} \ u_{n-1} \end{array}\right)$$
for all $n \geq 1$, and so,
$$\left(\begin{array}{c} u_{n+1} \ u_{n} \end{array}\right)=\left(\begin{array}{ll} 1 & 1 \ 1 & 0 \end{array}\right)^{n}\left(\begin{array}{l} u_{1} \ u_{0} \end{array}\right)$$
for all $n \geq 0$. Now, the matrix
$$A=\left(\begin{array}{ll} 1 & 1 \ 1 & 0 \end{array}\right)$$
has characteristic polynomial, $\lambda^{2}-\lambda-1$, which has two real roots
$$\lambda=\frac{1 \pm \sqrt{5}}{2} .$$

## 数学代写|离散数学代写DISCRETE MATHEMATICS代写|Generalized Lucas Sequences and Mersenne Primes

We just studied some properties of the sequences arising from the recurrence relation
$$u_{n+2}=u_{n+1}+u_{n} .$$
Lucas investigated the properties of the more general recurrence relation
$$u_{n+2}=P u_{n+1}-Q u_{n},$$
where $P, Q \in \mathbb{Z}$ are any integers with $P^{2}-4 Q \neq 0$, in two seminal papers published in 1878. Lucas numbers play a crucial role in testing the primality of certain numbers of the form $N=2^{p}-1$, called Mersenne numbers. A Mersenne number which is prime is called a Mersenne prime. We will discuss methods due to Lucas and Lehmer for testing the primality of Mersenne numbers later in this section.

We can prove some of the basic results about these Lucas sequences quite easily using the matrix method that we used before. The recurrence relation
$$u_{n+2}=P u_{n+1}-Q u_{n}$$
yields the recurrence
$$\left(\begin{array}{c} u_{n+1} \ u_{n} \end{array}\right)=\left(\begin{array}{cc} P & -Q \ 1 & 0 \end{array}\right)\left(\begin{array}{c} u_{n} \ u_{n-1} \end{array}\right)$$
for all $n \geq 1$, and so,
$$\left(\begin{array}{c} u_{n+1} \ u_{n} \end{array}\right)=\left(\begin{array}{cc} P & -Q \ 1 & 0 \end{array}\right)^{n}\left(\begin{array}{l} u_{1} \ u_{0} \end{array}\right)$$
for all $n \geq 0$. The matrix
$$A=\left(\begin{array}{cc} P & -Q \ 1 & 0 \end{array}\right)$$
has the characteristic polynomial $-(P-\lambda) \lambda+Q=\lambda^{2}-P \lambda+Q$, which has the discriminant $D=P^{2}-4 Q$. If we assume that $P^{2}-4 Q \neq 0$, the polynomial $\lambda^{2}-$ $P \lambda+Q$ has two distinct roots:
$$\alpha=\frac{P+\sqrt{D}}{2}, \quad \beta=\frac{P-\sqrt{D}}{2} .$$

## 数学代写|离散数学代写DISCRETE MATHEMATICS代写| Fibonacci and Lucas Numbers; Mersenne Primes

$$u_{n+2}=u_{n+1}+u_{n}, n \geq 0$$

$$u_{n+2}=u_{n+1}+u_{n}$$

$$\left(u_{n+1} u_{n}\right)=\left(\begin{array}{lll} 1 & 11 & 0 \end{array}\right)\left(u_{n} u_{n-1}\right)$$

$$\left(u_{n+1} u_{n}\right)=\left(\begin{array}{lll} 1 & 1 & 0 \end{array}\right)^{n}\left(u_{1} u_{0}\right)$$

$$A=\left(\begin{array}{lll} 1 & 1 & 1 \end{array}\right)$$

$$\lambda=\frac{1 \pm \sqrt{5}}{2}$$

## 数学代写|离散数学代写DISCRETE MATHEMATICS代写| Generalized Lucas Sequences and Mersenne Primes

$$u_{n+2}=u_{n+1}+u_{n} .$$

$$u_{n+2}=P u_{n+1}-Q u_{n}$$

$$u_{n+2}=P u_{n+1}-Q u_{n}$$

$$\left(u_{n+1} u_{n}\right)=\left(\begin{array}{lll} P & -Q 1 & 0 \end{array}\right)\left(u_{n} u_{n-1}\right)$$

$$\left(\begin{array}{lll} u_{n+1} & u_{n} \end{array}\right)=\left(\begin{array}{lll} P & -Q 1 & 0 \end{array}\right)^{n}\left(u_{1} u_{0}\right)$$

$$A=\left(\begin{array}{lll} P & -Q 1 & 0 \end{array}\right)$$

$$\alpha=\frac{P+\sqrt{D}}{2}, \quad \beta=\frac{P-\sqrt{D}}{2}$$

## MATLAB代写

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