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 $L_1=1$.

The $n$th Lucas number $L_n$ is defined recursively, by

$$L_n = L_{n - 1} + L_{n - 2}$$

for $n > 1$, and $L_0 = 2,\; L_1 = 1$. There is a closed form expression, namely

$$ L_{n}=\varphi ^{n}+(1-\varphi )^{n}=\varphi ^{n}+(-\varphi )^{-n}=\left({1+{\sqrt {5}} \over 2}\right)^{n}+\left({1-{\sqrt {5}} \over 2}\right)^{n},$$

where the golden ratio $\varphi$ is equal to $\dfrac{1 + \sqrt{5}}{2}$.

The sequence of Lucas numbers is: $2,1,3,4,7,11,18,29,47,76,123,\dots$ (sequence A000032 in the OEIS).

Reference: Lucas number.

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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 $c\phi^n + d\widehat{\phi}^n$ is the closed form for sequence…
JenkinsMa
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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. no invocation of the closed form of either) $$…
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Show that Fibonacci and Lucas numbers satisfy the following equality for all n ≥ 2.

Fibonacci numbers F1, F2, F3, . . . are defined by the rule: F1 = F2 = 1 and Fk = Fk−2 + Fk−1 for k > 2. Lucas numbers L1, L2, L3, . . . are defined in a similar way by the rule: L1 = 1, L2 = 3 and Lk = Lk−2 + Lk−1 for k > 2. Show that Fibonacci and…
user188043
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Relationship of powers of Phi to Lucas Numbers

I was watching a Numberphile and the interviewee was explaining various attributes of Lucas Numbers and he made the statement about creating a sequence by starting with the Golden Ratio and raising it to powers 1..n then rounding that number to…
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Are some Lucas numbers always coprime with all previous Lucas numbers?

I was looking at this webpage which lists the first 200 Lucas Numbers color-coded with their prime factors and I noticed that all the Lucas numbers with power of two or prime indexes were relatively prime to all previous Lucas numbers. Clearly the…
hatch22
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What is the Lucas counterpart to the Fibonacci identity $5F_n^2\pm~4=\lambda^2$?

It's a well-known rule that a number $x$ belongs to the Fibonnaci Sequence iff: $$\begin{align}5x^2\pm~4&=\lambda^2&\lambda\in\mathbb Z\end{align}$$ In other words, if and only if $5x^2\pm~4$ is a perfect square. What is the equivalent identity for…
Brian J. Fink
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Fibonacci/Lucas Number Congruences

Is there a compendium of well-known (and elementary) Fibonacci/Lucas Number congruences? I've proven the following and would like to know if it is (a) trivial, (b) well-known, or (c) possibly new. $$ L_{2n+1} \pm 5(F_{n+2} + 1) \equiv (-1)^{n} …
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$F_p,L_p$ 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 primes. It is also believed there are infinitely…
mick
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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 $$F_{2n+1} = F_nL_{n+1} + (-1)^n$$ are true. I did the base case $k =…
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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 induction? Does the second property follow from the…
Mary Star
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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 open problem how many Mersenne numbers are prime…
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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\…
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Proof about lucas numbers.

Define the Lucas numbers to be $$l_n = l_{n-1} + l_{n-2} $$ if $n \ge 2$ with initial conditions $l_0 = 2$ and $l_1= 1$. I "proved" by induction that $l_n = f_{n-1} + f_{n+1}$ for $n \ge 1$ (by $f_n$ I mean the n'th Fibonacci number) but never used…
BenHG
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Fibonacci and Lucas numbers related identities

We know that $H_n$ = $L_n + mF_n$, where $n = 0$ or $n > 0$ is simply relation between Fibonacci sequence and generalized Fibonacci-Lucas sequence. Are there any methods to prove the following identities? I am new to this site and hopefully, I will…
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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 Millin series $$ \sum_{n=1}^{\infty}…
choco_addicted
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