# Tagged Questions

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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### 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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### 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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### 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 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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### Finding index of a Fibonacci number: any mathematical solution possible?

The problem: Given a Fibonacci number,find its index. I am aware of the standard solution 'generate-hash-find'. I am just curious if there is ...
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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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### 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 ...
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### 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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### 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 ...
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### Is there a pattern of the length between one even Fibonacci number and another?

I had seen a math problem asking for the sum of all even Fibonacci numbers up to 4 million, but I still need to know this: Is there an obvious pattern of the distance between a even Fibonacci number ...
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### Why is the sum of any ten consecutive Fibonacci numbers always divisible by $11$?

I was wondering if anyone has any insights regarding the fact that the sum of any $a_1, \dots, a_{10}$ consecutive Fibonacci numbers is divisible by $11$ (and furthermore equals to $a_7*11$). What can ...
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### How do you determine if a number is a even Fibonacci number or not? [duplicate]

Rather than computing out the whole Fibonacci sequence and check if $n$ is even and in there, is there a more straightforward way to compute if $n$ is a even Fibonacci number?
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### Generalizing the Fibonacci sum $\sum_{n=0}^{\infty}\frac{F_n}{10^n} = \frac{10}{89}$

Given the Fibonacci, tribonacci, and tetranacci numbers, $$F_n = 0,1,1,2,3,5,8\dots$$ $$T_n = 0, 1, 1, 2, 4, 7, 13, 24,\dots$$ $$U_n = 0, 1, 1, 2, 4, 8, 15, 29, \dots$$ and so on, how do we show ...
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### Proving formulas with products of Fibonacci numbers

While digging through my old notes, I stumbled upon some formulas involving multiplication of Fibonacci numbers that I discovered about 7 years ago (being fascinated with Fibonacci numbers at the ...
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### proof by induction for golden ratio and fibonacci sequence

I have to prove the following equation by induction for $$x = \phi$$ I am stuck and I don't know how to proceed. This is the equation $$\phi ^n = f_n\phi + f_{n-1}$$ where $f_n$ is the nth term ...
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### Prove equality of Fibonacci sequence

Let $u(n)$ — the Fibonacci sequence. Prove that $$u(1)^3+...+u(n)^3=\frac{ 1 }{ 10 } \left[ u(3n+2)+(-1)^{n+1}6u(n-1)+5 \right].$$ I suppose we need prove that equality by iduction in $n$. ...
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### Period of Fibonacci mod $b$?

It is not too difficult to show that the Fibonacci numbers mod $b$ form a periodic sequence. I would like to say something interesting about the period. There is a small shortcut to the brute-force ...
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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 ...