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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How to prove that the Lucas Numbers satisfy these three properties [closed]

Prove these three properties: $L_1+L_3+L_5+\cdots+L_{2n-1} = L_{2n-2}-2$, $n\geq1$ $L_1^2 +L_2^2 +L_3^2+\cdots+L_n^2=L_nL_{n+1}-2$, $n\geq1$ $L_{n+1}^2 - L_n^2 = L_{n-1}L_{n+2}$, $n\geq2$ where $...
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1answer
135 views

Prove that if prime $p$ divide $a_{2k}-2$, then $p$ divide also $a_{2k+1}-1$.

Sequence $a_0,a_1,a_2,...$ satisfies that $a_0=2,a_1=1,a_{n+1}=a_n+a_{n-1}$ Prove that if $p$ is a prime divisor of $a_{2k}-2$,then $p$ is also a prime divisor of $a_{2k+1}-1$ If $x_{1,2}={1\pm\sqrt{...
0
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1answer
90 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 ...
2
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1answer
59 views

Lucas Number Sequence Theorem

How can I prove the following theorem using Induction about Lucas Numbers: $$a_{2n} = a_{n} \, b_{n}$$ Here, $a_{n}$ is the Fibonacci Sequence and $b_{n}$ is the Lucas Sequence. I tried to prove it ...
4
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1answer
148 views

Evaluate $\sum _{k=0}^{\infty } \frac{L_{2 k+1}}{(2 k+1)^2 \binom{2 k}{k}}$

How to prove $$\sum _{k=0}^{\infty } \frac{L_{2 k+1}}{(2 k+1)^2 \binom{2 k}{k}}=\frac{8}{5} \left(C-\frac{1}{8} \pi \log \left(\frac{\sqrt{50-22 \sqrt{5}}+10}{10-\sqrt{50-22 \sqrt{5}}}\right)\right)$$...
1
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19 views

How can I verify the ratio of Lucas numbers to fibonacci numbers algebreically

The article: "The Lucas numbers 1,3,4...are the sums of alternate Fibonacci numbers. The ratios of Lucas to Fibonacci must satisfy: $R_j = \frac{F_{i+1}+F_{i-1}}{F_i}=\frac{2F_{i+1}}{F_i-1}$ I ...
2
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1answer
90 views

Lucas sequence equivalent for the tribonacci sequence?

The Fibonacci and Lucas sequences occur within each other's identities, i.e. $$F_{2n} = F_{n} * (F_{n-1} + F_{n+1})$$ $$L_{n} = F_{n-1} + F_{n+1}$$ $$F_{2n} = F_{n} * L_{n}$$ The Lucas sequence ...
1
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0answers
35 views

if $p\mid u_m$, $m\mid n$, $p\mid u_n/u_m$, prove that $p\mid n/m$

If we have that $p\mid u_a$, $b\mid a$, and $p\mid u_a/u_b$, prove that $p\mid n/b$, assuming that $u_a$ and $u_b$ are terms in the linear recurrence for the Lucas Sequence. I've tried looking at the ...
4
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2answers
620 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 +...
1
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1answer
78 views

How to show that : $4(-1)^nL_n^2+L_{4n}-L_n^4=2$

How can we prove that: $$4(-1)^nL_n^2+L_{4n}-L_n^4=2$$ Where $L_n$ is Lucas number We got $L_n=\phi^n+(-\phi)^{-n}$ $4(-1)^nL_n^2=8(-1)^n\phi^{2n}+8$ $L_{4n}=\phi^{4n}+(-\phi)^{-4n}$ $L_n^4=4\phi^...
2
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0answers
51 views

Extra strong Lucas pseudoprimes and Jacobi symbol

In order to decide out whether a number $n$ is extra strong Lucas pseudoprime, one usually chooses Lucas sequence where Jacobi symbol $(D/n) = -1$. Such a $D$ can be found by Method C by Robert ...
3
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1answer
87 views

$F_p,L_p$ both prime?

It is well known that if the $n$ th Fibonacci number is a prime then it follows $n$ must also be a prime. So we wonder if $F_p $ is prime or not. It is believed there are infinitely many Fibonacci ...
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2answers
169 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 ...
2
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0answers
25 views

Different definitions of Lucas groups

I see two different definitions of Lucas groups, stated below. Is one of the two standard? Are they trivial variations? From these slides (Liljana Babinkostova et al., Boise State University, 2017) ...
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1answer
99 views

Lucas Numbers $(L_n)^2 = L_{2n} \pm 2$

When I was looking at the Lucas Number Series I noticed the following: If $n$ is odd, then $(L(n))^2 = L(2n) - 2 $ If $n$ is even, then $(L(n))^2 = L(2n) + 2 $ Can anyone provide a proof for why ...
2
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2answers
140 views

How to do Lucas Probabilistic Primality Test

I am trying to follow the steps to the Lucas Probabilistic Primality test, given on 83 of The Federal Information Processing Standards Publication Series of the National Institute of Standards and ...
2
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2answers
142 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 ...
3
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1answer
225 views

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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187 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. It referenced a paper by Mark Renault in 1996, with the quote from OEIS: http://webspace.ship.edu/msrenault/...
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1answer
119 views

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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1answer
77 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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1answer
124 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}{...
2
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1answer
179 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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3answers
103 views

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 ...
3
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2answers
144 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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0answers
135 views

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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0answers
44 views

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 $a^{p-1}\equiv1\pmod p$. ...
7
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0answers
509 views

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 ...
2
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2answers
93 views

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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4answers
1k views

Reccurence relation: Lucas sequence

I need to solve the given recurrence relation:$$L_n = L_{n-1} + L_{n-2},$$ $n\geq3$ and $ L_1 = 1, L_2 =3$ I'm confused as to what $n\geq3$ is doing there, since $L_1$ and $L_2$ are given I got $...
1
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1answer
263 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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0answers
118 views

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 ...
9
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1answer
135 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{...
2
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2answers
177 views

Fibonacci and Lucas series technique

Well, I have the following two problems involving Fibonacci sequences and Lucas numbers. I know that they share the same technique, but I don't have clear the procedure: $$f_n = f_{n-1} + f_{n-2}: ...
4
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2answers
324 views

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 ...
3
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1answer
207 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^...
2
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1answer
136 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?
4
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1answer
177 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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0answers
59 views

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 $\beta=1-\sqrt{2}$, ...
7
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4answers
550 views

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 ...
1
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2answers
51 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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0answers
92 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 $(W_n)_{n\geq 1}$...
1
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1answer
93 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{...
0
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1answer
121 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_{...
0
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1answer
136 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 $...
0
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1answer
78 views

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 ...
0
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1answer
146 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 ...
0
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2answers
529 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 ...
0
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1answer
52 views

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....
0
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0answers
543 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 ...