# 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 $L_1=1$.

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

How to prove $$\sum _{k=0}^{\infty } \frac{L_{2 k+1}}{(2 k+1)^2 \binom{2 k}{k}}=\frac{8}{5} \left(C-\frac{1}{8} \pi \log \left(\frac{\sqrt{50-22 \sqrt{5}}+10}{10-\sqrt{50-22 \sqrt{5}}}\right)\right)$$...
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### 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 ...
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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 ...
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### 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 ...
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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 ...
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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 ...
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### $5F_{n+1} = L_{n+4} − L_n.$

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 ...
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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) ...
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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 ...
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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 ...
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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 ...
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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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### Is there a proven way to calculate the entry point(first occurence) of a factor m, in the Fibonacci sequence?

I saw a comment at the OEIS website for the sequence of entry points of Fibonacci factors. It referenced a paper by Mark Renault in 1996, with the quote from OEIS: http://webspace.ship.edu/msrenault/...
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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 $$\sum_{i=0}^nF_i^2=F_nF_{n+1}.$$ ...
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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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### 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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### 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 ...
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### 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 ...
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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 ...
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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, $$3+\sqrt{11+\sqrt{11+\sqrt{... 2answers 177 views ### Fibonacci and Lucas series technique Well, I have the following two problems involving Fibonacci sequences and Lucas numbers. I know that they share the same technique, but I don't have clear the procedure:$$f_n = f_{n-1} + f_{n-2}: ...
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 ...