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 $L_1=1$.

Filter by
Sorted by
Tagged with
2
votes
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 ...
1
vote
0answers
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
votes
1answer
87 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
vote
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 ...
2
votes
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 ...
1
vote
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^...
3
votes
1answer
86 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 ...
4
votes
1answer
148 views

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

How to prove $$\sum _{k=0}^{\infty } \frac{L_{2 k+1}}{(2 k+1)^2 \binom{2 k}{k}}=\frac{8}{5} \left(C-\frac{1}{8} \pi \log \left(\frac{\sqrt{50-22 \sqrt{5}}+10}{10-\sqrt{50-22 \sqrt{5}}}\right)\right)$$...
2
votes
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) ...
0
votes
1answer
96 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
votes
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 ...
3
votes
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} = ...
0
votes
1answer
89 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 ...
1
vote
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 ...
1
vote
1answer
123 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}{...
1
vote
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. \...
2
votes
1answer
178 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} ...
1
vote
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
votes
2answers
143 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 ...
3
votes
1answer
222 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 ...
1
vote
0answers
133 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 ...
1
vote
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 $a^{p-1}\equiv1\pmod p$. ...
2
votes
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_{...
1
vote
1answer
262 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 ...
2
votes
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
votes
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{...
4
votes
2answers
314 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 ...
2
votes
2answers
141 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
votes
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^...
7
votes
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
votes
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?
5
votes
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{...
1
vote
0answers
57 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 $\beta=1-\sqrt{2}$, ...
1
vote
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 ...
0
votes
0answers
91 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
vote
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
votes
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
votes
1answer
134 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
votes
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 ...
4
votes
2answers
609 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 +...
0
votes
1answer
145 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
votes
2answers
520 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
votes
0answers
539 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 ...
0
votes
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....
1
vote
1answer
58 views

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 ...
0
votes
1answer
160 views

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 ...
7
votes
4answers
542 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
vote
1answer
180 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 ...
4
votes
1answer
174 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. ...
0
votes
0answers
329 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 ...