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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Periods of Fibonacci numbers in mod. Number Theory

Work out the periods $π(n)$ of the $\mod n$ such that $$f_k ≡ f_{k+π(n)} \mod n$$ I got $π(2)=3$,$π(3)=8$,$π(4)=6$ by computing it and looking at the periods. Now Part 2 Prove that for all ...
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159 views

Prove that $2^{n-1}$ divides $\binom{n}{1} + \binom{n}{3}5 + \binom{n}{5}25 + \binom{n}{7}125 + \cdots$ for $n \geqslant 1$.

Prove that $2^{n-1}$ divides $\binom{n}{1} + \binom{n}{3}5 + \binom{n}{5}25 + \binom{n}{7}125 + \cdots$ for $n \geqslant 1$. Assume $\binom{n}{k} = 0$ if $k>n$. Does anyone know an elementary ...
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2answers
50 views

Using induction to prove a formula for the Fibonacci sequence involving the solutions of $x^2=x+1$

Let $\{f(n)\}_{n=1}^{\infty}$ denote the Fibonacci sequence defined by $f(1)=1, f(2)=1$, and $f(n)=f(n-1)+ f(n-2)$ for all $n\geq 3$. Let $α=\dfrac{1+\sqrt{5}}{2}$ and $β=\dfrac{1-\sqrt{5}}{2}.$ ...
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1answer
42 views

Proving Fibonacci sequence with mathematical induction

Okay, so I have the following thing: $$\sum_{i=1}^a F_{2i}=F_{2a+1}-1 $$ It's to do with Fibonacci sequence. I can do the basis step of MI fine (proving for $a = 1$) However the inductive step has ...
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1answer
70 views

Mathematical induction proof of $\sum_{i = 1}^{n} F_{2i} = F_{2n + 1} - 1$

Use Mathematical Induction to show that $$\sum\limits_{i=1}^n F_{2i}=F_{2n+1}-1$$ for all integer $n\geq1$. My answer: Base case: for n = 1 $$\sum_{i = 1}^{n} F_{2i} = \sum_{i = 1}^{1} F_{2i} = ...
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Proof on Fibonacci sequence: $F(1) + F(3) + \cdots + F(2n-1) = F(2n)$ using induction and recursion

The problem is: Use induction and the recursive formula to prove that: $$F(1) + F(3) + \cdots + F(2n-1) = F(2n)$$ For the base case I let $n=1$ which gave $$F(1) = F(2(1))$$ $$1=1$$ Which is ...
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1answer
86 views

How to find pythagoras triplet using the fibonacci sequence?

I'm using the Fibonacci sequence to generate some Pythagorean triples $(3, 4, 5,$ etc$)$ based off this page:Formulas for generating Pythagorean triples starting at "Generalized Fibonacci Sequence". ...
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1answer
31 views

Fibonacci Sequence Squared

I have been learning about the Fibonacci Numbers and I have been given the task to research on it. I have been assigned to decribe the relationship between the photo (attached below). I know that the ...
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1answer
76 views

Consider a recursive sequence. Find all values of x for which this sequence is bounded

Consider a recursive sequence $a_{n+1} = a_n + a_{n-1}$ for all $n \geq 2$ with $a_{1} = 1$ and $a_2 = x$. Find all values of $x$ for which this sequence is bounded.
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1answer
62 views

Is there a formula to calculate factors of the smallest integer u, for which n, divides a Fibonacci number?

I have read that a conjecture for Fibonacci entry points, by Paul Bruckman and Peter Anderson has been proven for prime p, that uses the Galois theory and the Chebotarev density theorem to compute the ...
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0answers
60 views

Does the p-adic valuation of n, apply to Fibonacci numbers and Fibonacci-Wieferich primes?

Does the p-adic valuation of n, apply to Fibonacci-Wieferich primes in the following way? Let Fup be the smallest Fibonacci number divisible by a prime p > 5, then Vp(Fupk) = Vp(Fup) + Vp(k). The ...
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1answer
61 views

If $\frac{p_{n+1}}{np_n} \to p > 0 $, then $\sqrt[n+1]{p_{n+1}}-\sqrt[n]{p_{n}} \to \frac{p}{e}$

Problem: Prove that, if a sequence ${p_n}$ satisfies $p_n > 0$ and $\lim\limits_{n \to \infty} \frac{p_{n+1}}{np_n} = p > 0 $, then $\lim\limits_{n \to \infty} ...
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1answer
56 views

The sum of the Reciprocal of the Partial Sum of the Consecutive Fibonacci Numbers Series [closed]

How to prove that this conjecture is true for $n$th order Fibonacci number: $$1\le\sum_{n=1}^\infty\dfrac{1}{F_{n+2}-1}<2.5$$
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Calculating number of tile sequences

My daughter (aged 12) came to me with the problem below. I was able to help her to some extent but I could not see an age-appropriate solution. That is, I could imagine solutions involving factorials ...
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1answer
52 views

Are Fibonacci numbers with a square prime index always divisible by $F_p$?

I am doing some research on sequences and I need some help. The sequence of $F_{p^2}$ seems sort of different. It seems that because the index only has one distinct prime factor, as a result the only ...
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1answer
59 views

Showing $P_n = {F_n \over {2^n}}$

Let $P_n$ be the probability that, if you flip a fair coin $n$ times, there are no consecutive heads. Also, let $F_n$ be the $n^{th}$ Fibonacci number, normalized by $F_1 = 1$ and $F_2 = 2$ and $F_n = ...
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1answer
75 views

A Property of Fibonacci Numbers [duplicate]

I've seen the property $$f_{n+1} f_{n−1} = f_n^2 + (−1)^n, n ≥ 2.$$ for Fibonacci numbers at Abstract Algebra book of Thomas W. Judson. I've tried it for a few Fibonacci number, and I've really ...
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1answer
36 views

Fibonacci sequence developing [duplicate]

For the sum $$\sum_i^n {n-i \choose i}$$ I evaluate it for $n=1,2,3,4,5$ For $n=1$ we have $$\sum_{i=0}^1 {1-i \choose i} = {1 \choose 0} + {0 \choose 1} = 1 + 0 = 1$$ For $n=2$ we have ...
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1answer
49 views

Showing $F_{\frac{p^2+1}{2}}\equiv p-1 \pmod{p}$ when $p\equiv \pm 2 \pmod{5}$ and $p\equiv 3 \pmod{4}$

A while back I was messing around with representations of finite fields and found this problem above while doing so. I'll explain below how I came to this point but my question is: Question: How ...
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5answers
63 views

Proof about specific sum of Fibonacci numbers

Let $F_k$ denote the $k$-th Fibonacci number. Find a formula for and prove by induction that your formula is correct for all $n > 0$. $$ (-1)^0 F_0+(-1)^1 F_1+(-1)^2 F_2+\cdots+(-1)^n F_n=\ ? $$ I ...
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3answers
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Adding subscripts

This is a stupid question. But I'm trying to solve a Fibonacci problem and just realized that I don't know how to manipulate them. For example why does $F_{3n+1}$=$F_{3n-1}$+$F_{3n}$
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Fibonacci and Lucas congruence

I'd like to prove this congruence: $F_{2kt+n} \equiv (-1)^t F_n \pmod{L_k}$, where $F_n$ and $L_n$ are the Fibonacci and Lucas sequences. I have no idea how can i start. May anyone help?
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1answer
332 views

Prove by induction for $F(2n) = F(n)[F(n-1) + F(n+1)]$ for all $n\ge 1$

I am totally stumped by this question. I have proved the base case. Then for $k$ is $1$ assume the relation to be true. When I try to prove for $k+1$, the terms just do not simplify to what I want. Is ...
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1answer
45 views

Inequality involving Fibonacci numbers [closed]

If $F(n)$ are Fibonacci's numbers then prove that $$1< \frac{F(n+1)}{F(n)}<2$$ for all $n>2$
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On the Fibonacci sequence: is there an infinite number of primes $p$ dividing $F_{p-1}$?

Let $\{F_n\}_{n\geq 0}$ be the Fibonacci sequence. Prove that the number of primes $p$ so that $p\mid F_{p-1}$ is infinite. I tried to use induction, to no avail.
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1answer
49 views

A sequence related to squares of Fibonacci nubers

Let $f(n)$ be defined by $f(n)=f(n-1)+f(n-3)+f(n-4)$, for $n \ge 5$, $f(1)=1, f(2)=1, f(3)=2, f(4)=4$. First few terms of the sequence $(f(1), f(2), f(3), \ldots$) look like $(1, 1, 2, 4, 6, 9, ...
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2answers
269 views

Fibonacci sequence and eigenvalues

I'm learning about eigenvectors and values, and one of the excercises in my book tackles the fibonacci recursion from this angle. Let $F = \begin{bmatrix}1&1\\1&0\end{bmatrix}^n \quad\text{ ...
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Fibonacci infinite sum resulting in $\pi$

I found the following identity. While trying to prove it, I found some things that I don’t quite understand: $$\frac{\pi}{4}=\sqrt{5} \sum_{n=0}^{\infty} \frac{(-1)^n F_{2n+1}}{(2n+1) ...
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1answer
367 views

Infinite sum of reciprocal shifted Fibonacci numbers

I found on Wikipedia the following infinite sum : $$\sum_{k=0}^{\infty} \frac{1}{1+F_{2k+1}}=\frac{\sqrt{5}}{2}$$ There is no reference for this sum in the article and I couldn't find it ...
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80 views

Relationship between golden ratio powers and Fibonacci series

Can anyone prove the following equation? ($F_n$ is the $n$th element of Fibonacci series and $n \in N$.) $\phi = 1 \times \phi + 0$ $\phi^2 = 1 \times \phi + 1 $ $\phi^3 = 2 \times \phi + 1 $ ...
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Trying to prove $\sum_{i=1}^{N-2} F_i = F_N -2$

I'm trying to prove that $\sum_{i=1}^{N-2} F_i = F_N -2$. I was able to show the base case for when $N=3$ that it was true. Then for the inductive step I did: Assume $\sum_{i=1}^{N-2} F_i = F_N -2$ ...
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1answer
159 views

How to evaluate this infinite product ? (Fibonacci number)

Let $F_n$ be Fibonacci numbers. How to evaluate $$\prod_{n=2}^\infty \left(1-\frac{2}{F_{n+1}^2-F_{n-1}^2+1}\right)\text{ ?}$$ It seem like that $$\prod_{n=2}^\infty ...
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5answers
885 views

Fibonacci number identity.

How do I see that $f_{n+1}f_{n-1} = f_n^2 + (-1)^n$, $n \ge 2$, where $f_1 = 1$, $f_2 = 1$, and $f_{n+2} = f_{n+1} + f_n$ for $n \in \mathbb{N}$?
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Use induction to prove that $F_n \ge \sqrt 2 ^n$ for $n \ge 6$

The Fibonacci sequence $F_0, F_1, F_2,\dots$ is defined by the rule $F_0=0, \ F_1=1, \ F_n = F_{n−1} + F_{n−2}$. Use induction to prove that $F_n \ge \sqrt 2 ^n$ for $n \ge 6$. So for the base case: ...
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Prove that for every $k$ there exist fibonnaci number that ends with $k$ zeros.

Let $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n > 2$. Prove that for every $k$ there exist $F_m$ that ends with $k$ zeros. I tried using pigeonhole principle, but with no effect.
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Adding two variables with subscripts [closed]

What is the explanation to why $x_{3k} + x_{3k+1}$, is equal to $x_{3k+2}$. Isn't that incorrect because there is no value 1 in the subscript $x_{3k}$? I saw this in a prove in ...
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1answer
610 views

Parabolas in sequences of digits from the Fibonacci sequence

In preperation for an exam, I was studying Haskell. Therefore I was solving an old assignment where you had to define the fibonacci series. After solving the task (see 1] for source code) and ...
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1answer
374 views

Can the Fibonacci sequence be written as an explicit rule?

When I learned sums and sequences in algebra II with trig I learned about recursive rules and explicit rules. A recursive rule written with the formula of: $$a_n = r * a_{n-1}$$ Or as: $$a_n = ...
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Connections between Fibonacci and natural numbers

Here are some known facts about the Fibonacci numbers and then some questions regarding them . 1.Carmichael's theorem : For every $n>12$ $F_n$ has a prime divisor which doesn't divide any of ...
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1answer
237 views

Fibonacci numbers and the nontrivial zeros of the Riemann zeta function

Is this a mathematical coincidence? For $n=1,\dots,7$: $$ \left\lfloor \prod_{k=1}^n \arg\left(\rho_k\right)\right\rfloor = F_{n+1}, $$ where $\arg$ is the complex argument, $\rho_n$ is the $n$th ...
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1answer
189 views

Is there a proven formula for Pisano periods (Fibonacci numbers taken modulo $n$)?

I've seen on wikipedia this formula: $$\pi_k(p^n) = p^{n-1}\pi_k(p) $$ It says that the formula holds unless p is k-Wall-Sun-Sun prime, or k-Fibonacci-Wieferich prime, that is, $p^2$ divides ...
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Fibonacci proof question $\displaystyle \sum_{i=1}^nF_i = F_{n+2} - 1$

The sequence of numbers $F_n$ for $n \in N$ defined below are called the Fibonnaci numbers. $F_1 = F_2 = 1$, and for $n \geq 2$, $F_{n+1} = F_n + F_{n-1}$. Prove the following facts about the ...
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1answer
196 views

conjectured new generating function of fibonacci numbers

I conjecture a new generating function for the fibonacci numbers $F_{n}$. Given,the following conjectured q-continued fraction ...
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1answer
51 views

Let $a^n = a^{n - 1} + a^{n -2}$. Show that for any $A, B$, $F(n) = Aa^n + Bb^n$ satisfies Fibonacci recurrence relation.

$$\begin{align*} F(n) &= Aa^n + Bb^n\\ &= A(a^{n-1}+a^{n-2}) + B(b^{n-1}+b^{n-2}) \\ &= Aa^{n -1} + Aa^{n-2} + Bb^{n -1} + Bb^{n-2}\\ &= a^{n -1} (A + A^{a-1}) + b^{n - 2} (B + bB) ...
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1answer
40 views

Does the smallest prime factor of a Fibonacci number appear in the Fibonacci sequence?

I thought of a way to tackle the problem of knowing whether there are infinitely many Fibonacci primes or not and this question came to my mind: does the smallest prime factor of any Fibonacci number ...
2
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1answer
78 views

strange fibonacci recurrence

As it is well known fibonacci numbers satisfy the recurrence relation $$F_{n}=F_{n-1}+F_{n-2}$$ with initial conditions $F_{0}=0$ and $F_{1}=1$. While playing around with numbers,I noticed the ...
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3answers
74 views

How to prove a specific sequence is Fibonacci's with no prior knowledge nor trial and error?

Let $n$ be a positive integer and let $s_n$ be the number of increasing sequences of integers, alternatingly even and odd, starting with $0$ and ending with $n$. E.g. for $n=3$ we only have the two ...
7
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1answer
96 views

Closed form of series involving Fibonacci numbers

Let $F_n$ denote the $n$-th Fibonacci number and $\phi$ be the golden ratio, that $\phi = \frac{1+\sqrt{5}}{2}$. Find a closed form for the sum: $$\sum_{n=0}^{\infty} \frac{1}{(5\phi)^n(n+2)} ...
6
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3answers
1k views

People sitting in a circle chewing gum

Ten people are sitting in a circle of ten chairs, chewing gum. Each person spits out his or her gum and places it either under his or her own chair or under an immediately adjacent chair. How many ...
2
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2answers
119 views

Show by induction that $F_n \geq 2^{0.5 \cdot n}$, for $n \geq 6$

I have the following problem: Show by induction that $F_n \geq 2^{0.5 \cdot n}$, for $n \geq 6$ Where $F_n$ is the $nth$ Fibonacci number. Proof Basis $n = 6$. $F_6 = 8 \geq 2^{0.5 \cdot ...