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Questions tagged [lucas-numbers]

Questions on the Lucas numbers, a special sequence of integers that satisfy the recurrence $L_n=L_{n-1}+L_{n-2}$ with the initial conditions $L_0=2$ and $L_1=1$.

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fibonacci and lucas numbers induction

I'm having trouble proving by induction that this following Fibonacci-Lucas equation $$F_{2n+k} = F_n L_{n+k} + (-1)^n F_k \tag{*}$$ is true, given that $$F_{2n} = F_nL_n$$ and $$F_{2n+1} = ...
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34 views

Decomposing Fibonacci Numbers

This link demonstrates certain decompositions of Fibonacci numbers into products and sums of smaller Fibonacci numbers, such as $F_{m+n} = F_{m-1}F_n+F_mF_{n+1}$. I am wondering if anyone knows of ...
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115 views

$5F_{n+1} = L_{n+4} − L_n.$

I'm very new to induction proof and need some help to show that for $n ∈ N$ we have the relation between the Fibonacci and Lucas numbers: $$5F_{n+1} = L_{n+4} − L_n.$$ I know that I should show true ...
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1answer
84 views

Lucas number identity

Let $L_n$ be the Lucas numbers, defined by the recursion $L_n=L_{n-1}+L_{n-2}$ with initial values $L_0=2$ and $L_1=1$. Any idea how to prove the identity $$\sum_{j\ge{0}}(-1)^{n-j}\left(\binom{2n}{...
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47 views

Show that $F_{3n} = F_{n}(L_{2n} + (-1)^n)$

Let $F_n, L_n$ be the Fibonacci and Lucas sequences respectively. Show that $F_{3n} = F_{n}(L_{2n} + (-1)^n)$. In my attempt I am using Binet's formula, and the equivalent for the Lucas numbers. \...
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83 views

Proof by Induction of Sum of Squares of Fibonacci using Difference Opperators

Consider the sequence of Fibonacci numbers $\{F_n\}_{n\geq0}$ where $F_0=0,F_1=1$ and $F_n=F_{n-1}+F_{n-2}$ for $n\geq2$. It is proved that \begin{equation}\sum_{i=0}^nF_i^2=F_nF_{n+1}.\end{equation} ...
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Proof of a Well-Known Fibonacci Identity Involving Cubes of Fibonacci Numbers

The following is due to Lucas in 1876: $$F_{n + 1}^3 + F_n^3 - F_{n - 1}^3 = F_{3n}$$ I am unable to locate an elementary proof of this identity, and am unable to reproduce it myself. Would anyone ...
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74 views

Properties of Lucas sequence

I want to prove the following properties of Lucas sequence: $3\mid L_m \iff m\equiv 2\pmod 4$ $L_k\equiv 3\pmod 4$, where $2\mid k$ and $3\nmid k$. $$$$ For the first property do we use ...
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Does some Lucas sequence contain infinitely many primes?

Does some nontrivial Lucas sequence contain infinitely many primes? The Mersenne numbers $M_n=2^n-1:n$ not necessarily prime are a Lucas sequence with recurrence relation $x_{n+1}=2x_n+1$. It's an ...
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Lucas Number Non-Residues

I am currently reading the proof that if $L_n$ is two times a square, then $n = 0, \pm 6$. Throughout the paper containing this proof, there are several references to non-residues of Lucas and ...
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How to construct spiral phyllotactic pattern with the given number of spirals?

It is known that the spiral phyllotactic pattern is common in Nature, especially in Botany. It consists of two group of clockwise and anticlockwise spirals, starting from the center. In most cases ...
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Lucas Reciprocity Laws

Suppose $p$, $q$ are primes such that $p=qk+1$. If $a$ is not $0$, $1,$ or $-1$, then $a^q\equiv1\pmod p$ if and only if $a$ is a $k$-th power residue modulo $p$, so that $a^{p-1}\equiv1\pmod p$. ...
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Without resorting to induction show that $L_n^2=L_{n+1}L_{n-1}+5(-1)^n$,Where $L_n$ is $n^{th}$ Lucas number.

Without resorting to induction show that $L_n^2=L_{n+1}L_{n-1}+5(-1)^n$,Where $L_n$ is $n^{th}$ Lucas number. By definition of Lucas number $L_n=L_{n-1}+L_{n-2}\implies L_{n-1}=L_n-L_{n-2}$ and $L_{...
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135 views

Lucas Sequence and primality tests. is this test deterministic?

consider lucas parameters $(P, Q)$ and $D = P^2 - 4Q$. Let $n>0$,$\big(\frac{D}{n}\big)= - 1$ then $U_{n + 1}\equiv{0 \pmod{n}}$ and $n$ is a Lucas probable prime. This test base only on the ...
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Is this the best primality test using second order recurrences (Lucas Sequences)?

little Explanation Using second order lucas sequences $$U_{n + 2} = P\cdot{U_{n -1}} - Q\cdot{U_{n}}\qquad U_0=0, U_1=1$$ $$V_{n + 2} = P\cdot{V_{n -1}} - Q\cdot{V_{n}},\qquad V_0=2, V_1=P$$ Now our ...
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1answer
104 views

On $3+\sqrt{11+\sqrt{11+\sqrt{11+\sqrt{11+\dots}}}}=\phi^4$ and friends

Let $\phi$ be the golden ratio. We know it has a beautiful infinite nested radical, $$\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\dots}}}}=\phi$$ However, it is also the case that, $$3+\sqrt{11+\sqrt{11+\sqrt{...
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Mathematical induction on Lucas sequence and Fibonacci sequence

I'm trying to prove the following: $$L_k^2-5F_k^2=4(-1)^k\qquad k\ge1$$ $L_k$ is the $k$th term of the Lucas numbers and $F_k$ is the $k$th term of the Fibonacci sequence. I've tried using ...
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70 views

What do we know about Lucas sequence entry points?

For Lucas sequences Un(P, Q); X0=0; X1=1; Xn = P * Xn-1 - Q * Xn-2 Z(n) being the entry point of the sequence, which is the index of the first term divisible by n. What do we know about z(n)? Is ...
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194 views

Is there a polynomial mod $p$?

let $p$ be a fixed prime ($p\neq2,3,5$). Then, is there an even polynomial $f(x)$ with $deg(f)=p-5$ which satisfies the following equality? if $p\equiv1,4\ (mod\ 5)$ $1+x^2-x^{p-1}-x^{p+1}\equiv(x^...
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Period of Fibonacci sequence and Lucas number mod p

Let $p$ be an odd prime and $L_n$ be the $n$th Lucas number. Can anyone prove this? $$\frac{L_1}{1}+\frac{L_3}{3}+\frac{L_5}{5}+\cdots+\frac{L_{p-2}}{p-2}\neq0\pmod{p}$$ Please help me! I am ...
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1answer
111 views

How to prove this Fibonacci identity? $\sum_{k=0}^{n} F_{k} F_{n-k} = \frac{1}{5}\left(n L_{n} - F_{n}\right)$ [closed]

How to prove this Fibonacci identity? $$\sum_{k=0}^{n-3} F_{k} F_{n-k-3} = \frac{(n-3)L_{n-3} - F_{n-3}}{5}$$ i tried to used the generating function and partial decomposition but i got confused?
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Pell-Lucas number

I'm studying about Pell number and Pell-Lucas number whose have Binet formula $P_n=\frac{\alpha^n-\beta^n}{\alpha-\beta}$ and $Q_n=\alpha^n+\beta^n$, where $\alpha=1+\sqrt{2}$ and $\beta=1-\sqrt{2}$, ...
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2answers
44 views

Gaussian primes and Lucas numbers

Let $L_{n}$ be the $n$ th Lucas number. For example, $L_{1} = 1, L_{2} = 3, L_{3} = 4$. Conjecture: there is no Gaussian primes in the sequence $(L_{n-1} + L_{n} i)$ for $n = 2$ to $\infty$. I hope ...
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70 views

Lucas sequence are elliptic sequence?

I'm studying Elliptic Curves and EDS (Elliptic divisibility sequences) and working on Silvermans exercises 3.34 in "The arithmetic of elliptic curves": "An EDS over $K$ is a Sequence $(W_n)_{n\geq 1}$...
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81 views

Fibonacci and Lucas linear relation proofs

I really need some help in doing this: By using the generating functions $F(z)$ and $L(z)$ for Fibonacci and Lucas numbers, show that: $$ F_n = \frac{L_{n-1}}{2}+\frac{L_{n-2}}{2^2}+\ldots+\frac{...
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105 views

Prove that $L_{6k} \equiv 2$ (mod $4$)

Here is my reasoning so far $L_{6k} =F_{6k-1} + F_{6k+1}$ I have proved that any $F_n$ with n a multiple of 3 is even i.e. $F_{3n}$ is even and so is $F_{6n}$, it follows that $F_{6k-1}$ and $F_{...
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68 views

Prove that if $n$ is not a multiple of $3$ then $\gcd(F_n,L_n)=1$

I have that $\gcd(F_n,L_n)= \gcd(F_n, 2F_{n-1})$ I also proved earlier that $F_{3n}$ is even but that does that mean that all Fibonacci numbers obey this. In other words if $n$ is not a multiple of $...
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Prove that $\gcd(F_{3K},L_{3k}) \equiv 2$

Following this definition $L_K = F_{K-1} + F_{K+1}$ We have that $\gcd(F_{3K},L_{3k}) = \gcd(F_{3k}, F_{3k+1} + F_{3k-1}) =\gcd(F_{3k}, 2F_{3k-1})$ I don't know where to go from here. How do I ...
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253 views

Lucas and Fibonacci Numbers

Problem: Let \begin{align*} A_0 &= 6 \\ A_1 &= 5 \\ A_n &= A_{n - 1} + A_{n - 2} \; \textrm{for} \; n \geq 2. \end{align*} There is a unique ordered pair $(c,d)$ such that $c\phi^n +...
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1answer
119 views

Lucas Number Questions!

Problem: Find $(a,b)$ such that $$L_n = a\phi^n + b\widehat{\phi}^n.$$ Where $n$ is the $n^{th}$ lucas number. How would I start this? Would I just start by plugging in $a=b=1$ and then ...
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302 views

Why does the fibonacci series start with 0 and the lucas series with 1?

Why the difference? And when we're deriving these series from eigenvectors, what difference does the starting point make? Please help. I'm very confused. I have a test tomorrow and need to know the ...
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395 views

Finding a closed formula for the nth Lucas Number

The Lucas numbers are defined by $$L_0 = 1, L_1=3$$ $$L_n = L_{n-2} + L_{n-1}$$ I used this knowledge to get an equation for the nth Lucas number as follows: $$L(x) = \frac{1+2x}{1-x-x^2}$$ Now I ...
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1answer
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Is there a proof for why the difference between the n-th power of phi and the n-th Lucas number converges to zero?

Let $\epsilon(n)$ be the absolute value of the difference between the $n$th Lucas number ($L(n)$) and the $n$th power of $\phi$. $\epsilon(n)$ pretty clearly converges to zero, and does so pretty fast....
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lucas numbers Prove that $l_0^2+l_1^2+…+l_n^2=l_n*l_n+1+2$ for $n \ge 0$

Suppose that the lucas numbers are $l_n=l_{n-1}+l_{n-2}$ for all $n \ge 1$ where $l_0=2$ and $l_1=1$. Prove that $l_0^2+l_1^2+...+l_n^2=l_n*l_{n+1}+2$ for $n \ge 0$ I think the easiest way to prove ...
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Lucas Numbers Matrix A

In linear algebra I have an equation $x_n = Ax_0$. I know the values of $x$ for any given value of $n$, and I know $x_0$. Both are $2\times 1$ matrices. How do I solve for $A$? The answer should ...
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Is there a Lucas-Lehmer equivalent test for primes of the form ${3^p-1 \over 2}$?

I'm reviewing the cyclotomic form $f_b(n)= {b^n-1 \over b-1}$ for various properties to extend an older treatize of mine on that form. With respect to primality there is the Lucas-Lehmer-test for ...
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1answer
162 views

Why are the Lucas numbers and Fibonacci numbers linearly independent?

The answer for this question states (without giving too much else away): Since $F$ and $L$ are linearly independent ... Using the definition of linear dependence for infinite dimensions, I presume ...
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117 views

Prove by induction that $ F_{2n}=F_nL_n $

In the following exercise from George E. Andrews' Number Theory, we are given that $F_n$ and $L_n$ represent the $nth$ Fibonacci and Lucas numbers respectfully, and we need to prove by induction (i.e. ...
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288 views

Proofing that the Lucas numbers come closer to the Phi rounded numbers then the Fibonacci numbers.

Morning everyone, Bit of background, I'm a mid level programmer with very limited mathematics skills. As part of an assessment for a new role I've been asked to complete a technical task which ...
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399 views

Proving the closed form of a generating function of the sum of n lucas numbers is equal to the n+2th lucas number

1760887     I've been working on this homework problem for a while now and can't seem to solve it. Let $L_n = L_{n-1} + L_{n-2}$ for $n\ge 2$ where $L_0 = 2$ and $L_1 = 1$ $M_n = 1 + \sum_{i=0}^n{...
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1answer
104 views

Show that $\sum_{k=1}^{\infty}\frac{F_{2^{k-1}}}{L_{2^k}-2}=\frac{15-\sqrt{5}}{10}$

Show that$$\sum_{k=1}^{\infty}\frac{F_{2^{k-1}}}{L_{2^k}-2}=\frac{15-\sqrt{5}}{10},$$ where $F_n$ is a Fibonacci number and $L_n$ is a Lucas number.$^1$ Motivation: For example, when calculating ...
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194 views

Proofs with Fibonacci and Lucas numbers via induction

How would I go about proving the following sequence using induction on $k$? $2F_{2n+k} = F_{n+k}L_n + F_nL_{n+k}$ I know I have to show that it's true for $k = 1$, but I can't even seem to be able ...
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1answer
121 views

Prove that $L_n = \alpha^n +\beta^n$ for all integers $n\geq 0$

Let $\alpha =\left(\frac{1+\sqrt{5}}{2}\right)$ and $\beta = \left(\frac{1-\sqrt{5}}{2}\right)$. Prove that $L_n = \alpha^n +\beta^n$ for all integers $n\geq 0$ where $L_n$ denotes the Lucas numbers. ...
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165 views

Is there a proven way to calculate the entry point(first occurence) of a factor m, in the Fibonacci sequence?

I saw a comment at the OEIS website for the sequence of entry points, of Fibonacci factors. https://oeis.org/A001177 It referenced a paper by Mark Renault in 1996, with the quote from OEIS: http://...
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1answer
129 views

Are any factors of Lucas numbers divisible by a Fibonacci number greater than three?

The congruence relation for Fibonacci and Lucas numbers is stated: If Fn > 3 is a Fibonacci number then no Lucas number is divisible by Fn. However, does this apply to the factors as well?
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1answer
129 views

Fibonacci and Lucas numbers congruence relation?

The wikipedia page for Lucas Numbers seems to suggest that if $F_n ≥ 5$ is a Fibonacci number then no Lucas number is divisible by $F_n$. Here is the link. However, the page does not give any ...
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63 views

Perfect Squares on Lucas Sequences

Let the $f(x) = x^2 -ax+b$ has a positive discriminant $D=a^2-4b$ and $k,l$ be its roots. Then $U_n = \frac{k^n-l^n}{k-l}$ and $V_n=k^n+l^n$. I would like to prove these 4 properties If $U_n$ is a ...
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1answer
196 views

Lucas Number Sequence

Can anyone help me in this question: Define $ (b_n)$ as $b_1= 1,b_n=a_{n+1} - a_n $ for $ n\ge 2$, where $ a_n $ is the Fibonnaci series. This sequence is known as the sequence of Lucas numbers. ...
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1answer
117 views

How to show that $(L_n,F_n) < 3$ (Lucas numbers and Fibonacci numbers)

While following the proof that no Fibonacci number is a perfect square larger than 144 (https://math.la.asu.edu/~checkman/SquareFibonacci.html) I stumbled in proving two of the elementary facts about ...
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1answer
133 views

What is wrong with the following argument involving Fibonacci and Lucas numbers?

The Lucas numbers $L_n$ are defined by the equations $L_1 = 1$, and $L_n = F_{n+1} + F_{n-1}$ for each $n \geq 2$. What is wrong with the following argument? Assuming $L_n = F_n$ for $n = 1,2,\cdots,...