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 Numbers and Tilings

Show that $f_{n-1} + L_n = 2f_{n}$. So we need to find a $2$ to $1$ correspondence. Set 1: Tilings an $n$-board. Set 2: Tiling of an $n-1$-board or tiling of an $n$-bracelet. So we need to decompose a tiling of an $n$-board to a tiling of an…
NebulousReveal
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Fibonacci and Lucas numbers congruence relation?

The wikipedia page for Lucas Numbers seems to suggest that if $F_n ≥ 5$ is a Fibonacci number then no Lucas number is divisible by $F_n$. Here is the link. However, the page does not give any sources cited for me to find a proof of this; I was…
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How to show that $(L_n,F_n) < 3$ (Lucas numbers and Fibonacci numbers)

While following the proof that no Fibonacci number is a perfect square larger than 144 (https://math.la.asu.edu/~checkman/SquareFibonacci.html) I stumbled in proving two of the elementary facts about Lucas and Fibonacci numbers. (Lucas numbers have…
Mark Fischler
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Proof recursion is a subset of Lucas numbers

I need to prove that the recursion $a_n=\frac{a_{n-1}^2+5}{a_{n-2}}$ for $a_0=2,a_1=3$ are the Lucas numbers with even index. I would like to use induction, but I got a fraction that I'm not sure how to simplify into the clean recursion for the…
rkai
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How to find a complementary subspace for the subspace of sequences where $u_{n+2}=u_{n} + u_{n+1}$, in $\mathbb{R}^\infty$?

Consider this subspace of $\mathbb{R}^\infty$ (sequences of real numbers): $U = \{\vec{u} = (u_1, u_2, ...) \in \mathbb{R}^\infty | u_{i+2} = u_i + u_{i+1}$ for all $i\}$ My question is: how can i find a complementary subspace $V$ of $U$? Presumably…
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Confusion on the original Lucas test

I am currently researching on all primality tests deriving from Lucas' original paper Théorie des Fonctions Numériques Simplement Périodiques, which is of course known for its great deal of confusion. Still, I managed to make my way trough basically…
Plasma Stark
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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 contains the identity $$L_{2n} = L_{n}^2 + 2(-1)^n$$ Is…
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Prove relation between Lucas and Fibonacci numbers using tilings

I struggle a lot with combinatorial proofs and was hoping for some help. I need to prove by strong induction that $L_n = F_{n-2} + F_n$ and how this shows that $L_n$ counts the tilings of the circular $n$-board with $1$- and $2$-tiles EDIT ANSWER:…
Atom
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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 Baillie, for example. My question is, what to do when…
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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) The Lucas roup for parameters $(D,N)$ is the…
fgrieu
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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 Technology. I found this document by searching and…
northerner
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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} Suppose we generalize the Fibonacci numbers so…
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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…
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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 current primality tests use the divisibility…
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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 there research out there of z(n) for these…