4
votes
2answers
41 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. ...
2
votes
2answers
110 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 ...
44
votes
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 ...
2
votes
1answer
72 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 $n$ ...
3
votes
1answer
113 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 ...
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 ...
4
votes
2answers
111 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
vote
1answer
102 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 ...
2
votes
3answers
96 views

Is it true that $5^k \mid f(5^k)$?

I guess if it is true that $5^k \mid f(5^k)$, where $f(n)$ denotes the $n$-th Fibonacci's number. I have tried to prove it by induction on $k$, but nothing. Have you got any ideas?
5
votes
1answer
89 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
105 views

Magic Squares with Lucas and Fibonacci Numbers

I am quite curious about can we construct magic squares using only Lucas and Fibonacci numbers(of course not repeating them? If yes, how can we construct them? And if not , what is the proof?
-1
votes
2answers
125 views

how to find nth term of different fibonacci series with golden ratio [duplicate]

what i know : if i want to find $Nth$ term of a fibonacci series like : 1 1 2 3 5 8 13 21 ....... then to find $6th$ term we use golden ratio ...
1
vote
0answers
90 views

I am trying to prove this problem by induction, how can can i prove the following?

I am given $$ F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}} $$ where, $\varphi = \frac{1 + \sqrt{5}}{2}$ and $\psi = \frac{1 - \sqrt{5}}{2}$ The textbook states that it's equal to the n-th Fibonacci ...
5
votes
1answer
531 views

How to prove that Fibonacci number is integer?

How to prove that formula for Fibonacci numbers are always integers, for all $n$: $$ F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}} $$ where, $\varphi = \frac{1 + \sqrt{5}}{2}$ and $\psi = \frac{1 - ...
11
votes
1answer
337 views

Why does $\frac{1 }{ 99989999}$ generate the Fibonacci sequence?

$\frac{1}{99989999} = 1.00010002000300050008001300210034005500890144... \times 10^{-8}$ (Link), which includes the Fibonacci sequence $(1\ 1\ 2\ 3\ 5\ 8\ 13\ 21\ 55\ 89\ 144\ \ldots )$ This is ...
0
votes
2answers
101 views

Proving that $fib(n) < (5/3)^n$ for $n \ge 1$ by induction

I know this has been shown before here but no post really answered my question. I had this problem given to me as an induction practice problem and I couldn't solve it without help. When I got the ...
2
votes
0answers
69 views

Lucas Number Equivalent of the Pisano Period?

I'm doing some work with the Pisano Period, and it's leading to also talk about the Lucas numbers $mod \ m$, and their period. Is there a specific notation already designated? Sticking with using a ...
1
vote
1answer
243 views

Pisano periods of fibonacci mod

The wikipedia article on Pisano periods utilises the Binet's formula and quadratic residues to find $f(n)$ such that $F_n=f(n) \pmod{p}$ where $p$ is a prime number and $F_n$ is a Fibonacci number. ...
0
votes
2answers
216 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 ...
1
vote
0answers
286 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 ...
6
votes
3answers
1k views

Show that $f(2n)= f(n+1)^2 - f(n-1)^2$

Let $f(n)=f(n-1)+f(n-2)$ be the Fibonacci sequence with $f(0)=0,f(1)=1$. Show that $$f(2n)= f(n+1)^2 - f(n-1)^2.$$ I have tried several different approaches to this problem. Both inducting from the ...
1
vote
2answers
167 views

Counting function for Fibonacci numbers

Are there some results about "Fibonacci-counting function" - the function counting the number of Fibonacci numbers less than or equal to some real number x?
0
votes
0answers
86 views

Fibonacci sequense, problem od division

How to show that $7\mid F_m\Longrightarrow 8\mid m$ and $4\mid F_m\Longrightarrow 6\mid m$, knowing that (I) Two consecutive terms in the Fibonacci sequence are relatively prime. (II) In ...
0
votes
2answers
89 views

How to prove that the Fibonacci sequence $7\mid U_m\Longrightarrow 8\mid m$ and $4\mid U_m\Longrightarrow 6\mid m$

How to prove that the Fibonacci sequence $$7\mid U_m\Longrightarrow 8\mid m$$ and $$4\mid U_m\Longrightarrow 6\mid m$$I was confused because there $\{ 4,7 \}$ in Fibonacci sequece
6
votes
1answer
234 views

Let $F_n$ be a Fibonacci number and $p$ a prime. Verify that for $p \le 61$, if $p\equiv\pm1 \pmod{5}$ then $p\mid F_{p-1}$

Define the Fibonacci entry point of $p$ to be the least integer $n$ such that $p\mid F_n$ So for example, for $p = 3$ - the Fibonacci entry point is $n = 4$ since $F_4 = 3$ and obviously $3\mid 3$. ...
0
votes
1answer
94 views

Two issues of Number Theory

Knowing in Fibonacci sequence$$u_n\mid u_m\Longleftrightarrow n\mid m$$ Question 1: In Fibonacci sequence, show that $$5\mid u_m\Longleftrightarrow 5\mid m$$ Show: $\Longrightarrow$ In ...
2
votes
3answers
142 views

Statement on the Fibonacci sequence

Question: Let $n,m,\in\mathbb{N^*}$ with $n>1$ and let $u_n$ denote the $n$-th term of the Fibonacci sequence, then $$u_{n+m}=u_{n-1}u_m+u_nu_{m+1}$$ I know these theorems: Two consecutive ...
6
votes
0answers
462 views

A conjecture about Lucas series

Let $L_n$ be the Lucas series: $L_0=2,L_1=1,L_n=L_{n-1}+L_{n-2}(n>1).$ $p$ is a prime number and $p\equiv3,7\pmod {20}$, hence $\exists x,y\in \mathbb Z:2p=x^2+5y^2.$ Is it true that ...
8
votes
7answers
389 views

Why is the Fibonacci ratio though a decreasing function, it is alternating and decreasing?

I tried to find the ratio of consecutive terms of the Fibonacci series and found that it is a decreasing function and it converges . I tried it though a small code piece in python so that I can have a ...
4
votes
4answers
170 views

Profinite and p-adic interpolation of Fibonacci numbers

On the topic of profinite integers $\hat{\bf Z}$ and Fibonacci numbers $F_n$, Lenstra says (here & here) For each profinite integer $s$, one can in a natural way define the $s$th Fibonacci ...
6
votes
1answer
260 views

Fibonacci numbers that are powers?

The Fibonacci sequence is: $$\left(f_n\right) = \left(0,1,1,2,3,5,8,13,21,34,55,89,144,\dots\right)$$ where we start with $0$ and $1$ and each term in the sequence is the sum of the two previous ...
3
votes
2answers
139 views

Another Fibonacci identity

Here's a problem that is leading me in circles. Consider the Fibonacci number $F_n$ defined by $F_0 = 0$, $F_1 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for all $n \geq 2$. Prove that $F_{2n-1} = F_{n}^2 ...
1
vote
0answers
118 views

The zeta-function of Fibonacci sequence?

I have been told that, for a given sequence ${N_r}$, its zeta-function is given by $exp(\Sigma _{r=1}^{\infty} N_rT^r/r)$. Since I have barely any experience with such a sum, I tried to find some ...
4
votes
4answers
256 views

Fibonacci Proof

Prove that: $$F_1F_2+F_2F_3+F_3F_4+\cdots+F_{2n-1}F_{2n}=F_{2n}^2$$ I set it up so: $$F^2(2k) + F(2k+1)F(2k+2) = F^2(2k+2)$$ I've tried: $$F(2k)^2 + F(2k+1)*F(2k+2) = ...
2
votes
2answers
150 views

Proof That the Ratio of Sucessive N-nacci Numbers Tends to 2.

For the paper that I'm working on involving N-nacci Recursion Formulas I need to prove that $$ \lim_ {n \to \infty}\lim_{a \to \infty} \left | \frac{f^{(n)}_{a+1}}{f^{(n)}_a} \right | = 2$$ To start ...
1
vote
4answers
139 views

Proving that $\alpha^{n-2}\leq F_n\leq \alpha^{n-1}$ for all $n\geq 1$, where $\alpha$ is the golden ratio

I got stuck on this exercise. It is Theorem 1.15 on page 14 of Robbins' Beginning Number Theory, 2nd edition. Theorem 1.15. $\alpha^{n-2}\leq F_n\leq \alpha^{n-1}$ for all $n\geq 1$. Proof: ...
3
votes
3answers
272 views

Is there a closed form equation for fibonacci(n) modulo m?

Basically I am curious if there's a direct way to calculate fibonacci(n) modulo m with a closed form formula so I don't have to bother with matrix exponentials.
4
votes
2answers
112 views

How many ways to reach $1$ from $n$ by doing $/13$ or $-7$?

How many ways to reach $1$ from $n$ by doing $/13$ or $-7$ ? (i.e., where $n$ is the starting value (positive integer) and $/13$ means division by $13$ and $-7$ means subtracting 7)? Let the number ...
1
vote
2answers
297 views

Primitive squareful Fibonacci numbers

In https://oeis.org/A065069 numbers $n$ such that Fibonacci($n$) is not squarefree, but for all proper divisors $k$ of $n$, Fibonacci($k$) is squarefree, are listed. OEIS gives a Mathematica program ...
1
vote
0answers
206 views

Fibonacci and Lucas relations

We know that $H_n$ = $L_n + mF_n$, where $n = 0$ or $n > 0$ is simply relation between Fibonacci sequence and generalized Fibonacci-Lucas sequence. Are there any methods to prove the following ...
-3
votes
2answers
315 views

Fibonacci numbers extended

I am so excited and enjoyed the both the proofs of my previous question on Fibonacci series. I am so interested and fascinating person on fib series/functions. I use to do some rough work in my ...
1
vote
2answers
215 views

Fibonacci function

Dear Professors and Mathematcians, Now, I am introducing Fibonacci sequence and function. Consider, $F(x)$ is a Fibonacci function and $f_n$ is Fibonacci sequence. For fixing the initial values by ...
7
votes
2answers
311 views

Fibonacci numbers of the form $5x^2+7$

Numerically I find the positive integer solution of the equation $F_n=5x^2+7$, where $F_n$ denotes the $n^\text{th}$ Fibonacci number, as $(n,x)=(16,14)$ and I guess that the only positive solution of ...
2
votes
1answer
129 views

Finding n in Fibonacci closed loop form

The nth term of the Fibonacci series is given by $F_{n}$=$\Big\lfloor\frac{\phi^{n}}{\sqrt{5}}+\frac{1}{2}\Big\rfloor$ How do you get the following expression for n from this? ...
3
votes
1answer
163 views

Fibonacci numbers moduli

I have made some observation on very interesting material on Fibonacci series. I need some help in proving them mathematically. We can observe that the periodicity of Fibonacci numbers modulo m, ...
11
votes
1answer
296 views

Why do the Fibonacci numbers recycle these formulas?

The Fibonacci numbers $F_n = 0, 1, 1, 2, 3, 5, 8, 13, \dots$ obey the following recurrence relations, $ \begin{aligned} &F_{n}-\;F_{n-1}-F_{n-2} = 0\\[1.5mm] &F_{n-1}^3-F_{n}^3-F_{n+1}^3 = ...
7
votes
2answers
275 views

Fibonacci numbers with largest prime factor appearing more than once

$F_6=2^3$ and $F_{12}=2^43^2$. Is there an $n>12$ such that $F_n=p^2k$ with $p$ prime and $k$ is $p$-smooth?
5
votes
2answers
225 views

Is an algebraic formula for the number of cyclic compositions of n known?

From Wikipedia: In January 2011, it was announced that Ono and Jan Hendrik Bruinier, of the Technische Universität Darmstadt, had developed a finite, algebraic formula determining the value of p(n) ...
3
votes
1answer
646 views

Understanding fibonacci ratio in plants

Recently, a 13year old kid has re-dicovered that there is a magic ratio for branching in plants. Following article describes his work in his own words. ...
2
votes
1answer
388 views

Prime power divisors of the fibonacci numbers

I came across a result that if $p^n \mid f_m$ for some $n\geq1$ then $p^{n+1} \mid f_{pm}$. I was wondering if this is true.