Questions on the Fibonacci numbers, a special sequence of integers that satisfy the recurrence $F_n=F_{n-1}+F_{n-2}$ with the initial conditions $F_0=0$ and $F_1=1$.

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71 views

Fibonaaci Recurrence

This is an interesting question where we are trying to solve another recursion which has same tree structure as the given recursion and also has term similarities Given Data in question ...
0
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1answer
25 views

Initial values appear from nothing

This answer says that any casual sequence of the kind $y_n = y_{n-1} + y_{n-2} + y_{n-3} + \ldots $ will stay constant-0 because $y_0$ is a sum of zeroes, so is $y_1$ and the rest of the sequence. I ...
28
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2answers
585 views

Very curious properties of ordered partitions relating to Fibonacci numbers

I came across some interesting propositions in some calculations I did and I was wondering if someone would be so kind as to provide some explanations of these phenomenon. We call an ordered ...
0
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1answer
42 views

Fibonacci Numbers in Nature

Supposedly the Fibonacci sequence appears naturally in nature, and my question is how, where and I guess why? I read that one way this is so is that it models the population of honey bees under ideal ...
0
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1answer
38 views

Generalized Fibonacci Sequence

I'm having trouble with a problem I encountered while studying Number Theory. This problem comes from the book Number Theory by George E. Andrews. It defines a generalized Fibonacci sequence $F_1$, ...
2
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2answers
100 views

Fibonacci number ending with given sequence of digits

Related to this question: For any given sequence of digits, does a Fibonacci number exist ending with such sequence? If not, it would be nice to find the smallest counterexample. (in other ...
4
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2answers
39 views

Proving Fibonacci inequality

I didn't see a question regarding this particular inequality, but I think that I have shown by induction that, for $n>1$. I am hoping someone can verify this proof. ...
9
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3answers
355 views

Significance of starting the Fibonacci sequence with 0, 1…

DISCLAIMER: I do not deal with in-depth mathematics on a daily basis as some of you may, so please pardon my ignorance or lack of coherence on this topic. QUESTION: What is the significance of ...
41
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4answers
3k views

Fibonacci number that ends with 2014 zeros?

This problem is giving me the hardest time: Prove or disprove that there is a Fibonacci number that ends with 2014 zeros. I tried mathematical induction (for stronger statement that claims that ...
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1answer
29 views

Order of convergence for the method of false position

I'm reading about the order of convergence of the method of false position and there is one tricky point in the proof I don't understand. The method itself for finding the minimum $x^*$ of a function ...
7
votes
1answer
108 views

An analogue of Hensel's lifting for Fibonacci numbers

In this question Oleg567 conjectured the following interesting fact about Fibonacci numbers: $$ k\mid F_n\quad\Longrightarrow\quad k^d\mid F_{k^{d-1}n} $$ that can be regarder as an analogue of the ...
0
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2answers
123 views

Fibonacci series, which is most pure mathematically? [closed]

There are various methods of generating the Fibonacci series. I'm going to list 3 of them in this question. Method 1: f[n] = f[n - 1] + f[n - 2] Method 2: (more ...
3
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1answer
54 views

Find a sequence

Find the function for the sequence $a_0 = 0, a_1 = 1$ and $a_{n}=a_{n+10}+a_n$ for all $n>0$.
4
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0answers
69 views

Can I use the equality $\phi^2=\phi+1$ without proving it?

I am looking at the following exercise: $$\text{ Show with induction,that the } i^{th} \text{ number Fibonacci satisfies the equality: } $$ $$F_i=\frac{\phi^i-\hat{\phi}^i}{\sqrt{5}}$$ where $\phi$ ...
2
votes
1answer
60 views

Source for relationship between $d$-ary Fibonacci numbers and generalized golden ratio?

I am not a mathematician (but a computer scientist) and stumbled across the following in the analysis of an algorithm (Berthold Vöcking: How Asymmetry Helps Load Balancing). The author gives Knuth: ...
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3answers
49 views

How to compute the nth number of a general Fibonacci sequence with matrix multiplication?

If we want to compute the nth Fibonacci number we just power the matrix $M = \left[\begin{array}{cc} 1 & 1 \\ 1 & 0 \end{array}\right]$ $n$ times and we get $M =\left[ \begin{array}{cc} ...
2
votes
1answer
69 views

The number of partitions of $n$ and the $n$th Fibonacci number.

I'm very sorry if this is a duplicate in any way. There's a lot of material out there on connections between these sequences so it's a possibility . . . Let $P_n$ be the number of partitions of $n$ ...
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1answer
60 views

Checking a formula works for many numbers [closed]

Hello StackExchange users. I have discovered a formula which works out any Fibonacci number using the formula to work out the value of any cell in Pascal's triangle. It uses the sigma notation which I ...
3
votes
1answer
110 views

Golden ratio, $n$-bonacci numbers, and radicals of the form $\sqrt[n]{\frac{1}{n-1}+\sqrt[n]{\frac{1}{n-1}+\sqrt[n]{\frac{1}{n-1}+\cdots}}}$

The following infinite nested radical $$\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}$$ is known to converge to $\phi=\displaystyle\frac{\sqrt{5}+1}{2}$. It is also known that the similar infinite ...
12
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4answers
342 views

Converting recursive equations into matrices

How do we convert recursive equations into matrix forms? For instance, consider this recursive equation(Fibonacci Series): $$F_n = F_{n-1} + F_{n-2}$$ And it comes out to be that the following that ...
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1answer
67 views

Fibonacci numbers $F_{n+3} + F_{n} = 2F_{n+2}$

Prove $F_{n+3} + F_{n} = 2F_{n+2}$ for any positive integer n. So What I did was this: fn+ fn+1 = fn+2 fn + fn+1 = fn+2 => fn+2 -fn+1 fn+1 + fn+2 = fn+3 then I subsituted into equation in ...
0
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2answers
211 views

find a general expression for the remainder when a prime divides a fibonacci.

I have primes of form $5k\pm1$. Consider the equation: $F_n=f(n)\pmod p$ where $F_n$ is the nth fibonacci number. Now given a c, how can i check whether or not there exists a solution for $f(n)=c ...
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5answers
1k views

What is the summation notation for the Fibonacci numbers?

I learned about summation notation the other day, and I'm looking for a way to write the Fibonacci numbers with it. What would it look like?
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2answers
54 views

Why is $\sum_{k=0}^{n} f(n,k) = F_{n+2}$?

If $f(n,k)$ is the number of $k$ size subsets of $[ n ] = { 1 , \ldots , n }$ which do not contain a pair of consecutive numbers, how can I show that $\sum_{k=0}^{n} f(n,k) = F_{n+2}$? ($F_{n}$ is the ...
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4answers
132 views

Fibonacci Calculation using a larger matrix

So the formula to generate the fibonacci sequence in matrix form is: $$ \begin{pmatrix} 1 & 1 \\ 1 & 0 \\ \end{pmatrix}^n = \begin{pmatrix} F_{n+1} & F_n \\ F_n & ...
2
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2answers
87 views

How do I apply the $\pm4$ part of the equation $5F_n^2\pm~4=L_n^2$ without knowing $n$?

I'm trying to test a great many numbers $a^3+b^3$ to see if any of them are Fibonacci using the formula $$a^3+b^3=F_n \iff 5(a^3+b^3)^2\pm~4=L_n^2$$ I want to make my search more efficient by having ...
11
votes
5answers
1k views

How to tell if a Fibonacci number has an even or odd index

Given only $F_n$, that is the $n$th term of the Fibonacci sequence, how can you tell if $n \equiv 1 \mod 2$ or $n \equiv 0 \mod 2$? I know you can use the Pisano period, however if $n \equiv 1$ or ...
33
votes
4answers
761 views

Conjecture: There's only one Fibonacci number that is the sum of two cubes

As the title says, I need help proving or disproving that there is only one Fibonacci number that's the sum of two (positive) cubes, $2$. I did a small brute force test with Fibonacci numbers below ...
0
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1answer
34 views

Base case in the Binet formula (Proof by strong induction)

I don't understand why the author says that the inductive formula does not apply until $u_3$. The formula does not depend on other terms in the Fibonacci sequence, so I thought I could just prove it ...
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2answers
69 views

How to establish this inequality without using induction?

Given the Fibonacci sequence $a_1 = 1$, $a_2 = 2$, $\ldots$, $a_{n+1} = a_n + a_{n-1} $ for $n \geq 2$, how to derive, without using induction, the inequality $$ a_n < (\frac{1+\sqrt{5}}{2})^n $$ ...
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0answers
274 views

Calculating Pisano periods for any integer

I recently stumbled across this SPOJ question: http://www.spoj.com/problems/PISANO/ The question is simple. Calculate the Pisano period of a number. After I researched my way through the web, I found ...
5
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1answer
87 views

Number of zeros in Fibonacci sequences mod $p$

We know that Fibonacci sequences are periodic in mod $m$. For example, for $p\equiv \pm 1 \pmod 5$ and $\pm 2 \pmod 5$ the periods for Fibonacci sequences modulo $p$ divide $p-1$ and $2p+2$ ...
3
votes
1answer
73 views

Fibonacci series mod a number

I'm trying to write a program with an input of numbers $n$ and $k$ (where $n<10^{20}$ and $k<10^9$), where I compute fib[n] % k. What is a good FAST way of computing this? I have read many ...
2
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1answer
351 views

Finding the binary representation of the $n$th Fibonacci term

Objective: To find the binary representation ( or no. of 1's in binary representation) of nth term in Fibonacci sequence where n is of the order 10^6. My current approach: Find nth term (in decimal) ...
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1answer
99 views

What is length of period of fibonacci number mod 1000033

Can some explain me how the period of Fibonacci mod $1000033$ is $4684$. As we know if $n$ mod $5$ is $2$ or $3$ then period is $2n + 2$ so the period should me $2\times1000033 + 2$ but why it is ...
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1answer
129 views

Asymptotic value of Fibonacci numbers

It is well known that $F_n\sim\frac{\phi^n}{\sqrt{5}}$, where $\phi=\frac{1+\sqrt{5}}{2}$. Does someone know a better estimate? With proof please. I'm trying to calculate the following limit: Let ...
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5answers
163 views

A Non-recursive Fibonacci Sequence

How can I determine the general term of a Fibonacci Sequence? Is it possible for any one to calculate F2013 and large numbers like this? Is there a general formula for the nth term of the Fibonacci ...
0
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1answer
72 views

Let $u_n$ be the $n$-th entry in the Fibonacci sequence $1,1,2,3,5,8,13,\ldots$

If you start with $u_1 = 1$ and $u_2 = 1$, then the sequence can be generated using the formula $$u_{n+1} = u_n + u_{n-1}\ .$$ If $u_n = r^n$, what is r? Can anyone figure this out? I am so stuck ...
4
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2answers
105 views

The sum of $n$ consecutive Fibonacci numbers.

The sum of $8$ consecutive Fibonacci numbers is divisible by $3$. How can I generalize this for the sum of $n$ consecutive Fibonacci numbers? For example, $$1+1+2+3+5+8+13+21=54=3\times 18 \\ ...
1
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1answer
58 views

Proof that golden angle successively divides the largest gap by the golden ratio?

The golden angle divides the circumference of a circle by the golden ratio. "If radial spokes are placed successively into the circle, each spaced by a golden angle increment, then each additional ...
28
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4answers
544 views
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4answers
86 views

Determine which Fibonacci numbers are even

(a) Determine which Fibonacci numbers are even. Use a form of mathematical induction to prove your conjecture. (b) Determine which Fibonacci numbers are divisible by 3. Use a form of mathematical ...
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4answers
46 views

Help understanding Recursive algorithm question

We have a function that is defined recursively by $f(0)=f_0$, $f(1)=f_1$ and $f(n+2) = f(n)+f(n+1)$ for $n\geq0$ For $n\geq0$, let $c(n)$ be the total number of additions for calculating $f(n)$ ...
2
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1answer
52 views

What's the Lucas version of the Möbius test for Fibonacci numbers?

I recently came across the following, attributed to Möbius: $$(a\in\mathbb N)=F_n\iff\left[\varphi a-\tfrac{1}{a},\varphi a+\tfrac{1}{a}\right]\ni(b\in\mathbb N)$$ It is the lesser-known test used to ...
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1answer
34 views

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 $5x^2\pm~4$ ...
1
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1answer
44 views

Looking for a sum 1/(Fib(n)*Fib(n+2))

I am looking for the sum of the series: $$\sum_{n=1}^{\infty}\frac{1}{F_{n}F_{n+2}},$$ where $F_{n}$ is the $n$-th Fibonacci number. I was thinking about splitting the fraction into 2 like in the ...
0
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0answers
29 views

How can i prove that result on Fibonacci and coprimes?

Let $n,q\in\mathbb{N^*}$. How can i show that $F_n\wedge F_q=F_{n\wedge q}$ ? (Or, is it false ?) ($a\wedge b = GDC(A,B)$) I have absolutely no idea how to do that ...
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2answers
56 views

Compute $F_{1000} \bmod F_{11}$, where $F_n$ denote the Fibonacci numbers

Compute $F_{1000} \bmod F_{11}$, where $F_n$ denote the Fibonacci numbers. Progress: $F_{11}=89$ . I believe you should find the period of $F_n \bmod 89$ and use that to solve it. But I'm not not ...
8
votes
2answers
777 views

Find all integer solutions for the equation $|5x^2 - y^2| = 4$

In a paper that I wrote as an undergraduate student, I conjectured that the only integer solutions to the equation $$|5x^2 - y^2| = 4$$ occur when $x$ is a Fibonacci number and $y$ is a Lucas number. ...
0
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1answer
31 views

finding $n$ consecutive composite Fibonacci numbers.

For each $n$, How can we find $n$ consecutive composite Fibonacci numbers?