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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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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Closed formula for Lucas numbers

Possible Duplicate: Prove this formula for the Fibonacci Sequence How does one find a formula for the recurrence relation $a_{1}=1,a_{2}=3, a_{n+2}=a_{n+1}+a_{n}?$ How do I go about obtaining a closed formula for Lucas numbers? The Lucas numbers…
soniccool
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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 perfect square then $n=1,2,3,6$ or $12$ If $V_n$ is…
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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. Prove: (i) $b_n=b_{n-1} + b_{n-2}$ for $ n \ge…
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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 $n = 1,2,\cdots,k$, we see $L_{k+1} = L_k + L_{k-1}$ from an…
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Solving a question about Fibonacci and Lucas numbers using induction

Im working on practice problems that the instructor gave us yesterday, and I absolutely have no clue of how to solve this problem.. I need to use mathmatical induction to solve this problem.. The question is: Fibonacci numbers F1, F2, F3, . . . are…
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How to find the count of edge covers of a graph of degree 2?

I have been trying to understand this editorial for question F of Atcoder beginner contest 247: https://atcoder.jp/contests/abc247/editorial/3773 . What I have not been able to understand is this part: How many subsets of edges in an M-vertex graph…
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Convert LUC formula to GF2 polynomial base

is it possible to convert LUC public key encryption formula from standard math to finite field GF(2) polynomial like RSA? LUC is based on Lucas function and it uses the following equation V(i+1)=mv(i)-Qv(i-1) mod (p*q) Q=1, V(0)=2, V(1)=m, m =…
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How can this relation between Lucas and Fibonacci numbers be proved?

$ \lim_{n \to \infty}\Bigg(\dfrac{2 \cdot 10^n + 1 \cdot 10^{n-1} + 3 \cdot 10^{n-2} + 4 \cdot 10^{n-3}+...}{0\cdot 10^n + 1 \cdot 10^{n-1} + 1 \cdot 10^{n-2} + 2 \cdot 10^{n-3}+...}\Bigg) = 19 $ I noticed that the left-hand side converges to $19$,…
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Does $(L_{p+1}+2)\equiv0 \mod p$ only when $p^2$ have digits in nondecreasing order?

Let $L_n$ be the $n$th Lucas number and $p$ a prime number. I noticed something with Lucas Number : it seems than $(L_{n+1}+2)\equiv0 \mod n$ is right only when $n$ is a prime $p$ and only if $p^2$ has digits in nondecreasing order. For example…
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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 this is always true?
Jojo197
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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 more general decompositions of Fibonacci numbers…
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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 $(W_n)_{n\geq 1}$ defined by four initial conditions $W_1, W_2,…
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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 $F_{6k+1}$ are odd and an odd $+$ and odd $=$ an…
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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 $3$ then the corresponding Fibonacci number is odd?…