# 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 $L_1=1$.

73 questions
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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