Questions tagged [fibonacci-numbers]
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52 questions
18 votes
2 answers
2k views
Probability of no triangle trios among $n$ random lengths in $[0,1]$: Alternative Proofs
In August 2025, SciAm ran a piece about an arXiv preprint, last revised May 2025, that proves the following: Select $n$ stick lengths independently and uniformly at random from the interval $[0, 1]$. ...
- 8,106
0 votes
1 answer
315 views
Fibonacci Diophantine Equation
Let $F_n$ denote the $n$th Fibonacci number and $a$ and $b$ nonzero coprime integers. Is it true that the DE $x^3 = aF_{n}+b$ has finitely many solutions?
- 1,205
2 votes
0 answers
143 views
Properties of OEIS sequence A061446 (primitive parts of Fibonacci numbers)
The first terms of A061446 are $(p(n))_{n\geq 1} =(1,1,2,3,5,4,13,7,17,11,89,6,233,29,\dots)$. I know that $p(n)=\phi(n,5)$ for $n\geq 2$, where $\phi(n,x)$ denotes the Minimal Polynomials of $(2sin(\...
- 5,832
3 votes
1 answer
246 views
$p$-adic valuation of sum of Fibonomial coefficients
Let $\binom{n}{k}_F$ be the fibonomial coefficient and $p$ a prime $\equiv 3, 7 \pmod{20}$ (OEIS A053027). Is it true that: $$\nu_p\Bigg(\sum_{k=0}^{(p+1)n+1} \binom{(p+1)n+1}{k}_F\Bigg) = \nu_p((2pn)!...
- 1,186
0 votes
1 answer
155 views
Closed-form formula for limit associated to non-standard random Fibonacci sequences
Let $X_{n+1} = U_n X_n + V_n X_{n-1}$ where $X_0 = X_1 =1$ and $(U_n), (V_n)$ are sequences of random variables with $E[U_n] = E[V_n] = 1$, both independent, identically distributed and independent ...
- 3,158
14 votes
1 answer
1k views
Can the Fibonacci condition in OEIS:A337625 be replaced by asking whether the number is a multiple of 5?
This question is motivated by the "pseudo-prime" question Primality test using the Golden Ratio . Consider the two sequences I. odd composite integers satisfying the two conditions $F_n^2\...
- 201k
2 votes
0 answers
145 views
Closed form expression for the following Polynomial
Does anyone recognize this recursive polynomial? $$ \rho_{i+1}(z)=\rho_0(z)+\sum_{j=0}^i(j+1) z \rho_{i-j}(z),\qquad \rho_0(z)=1 $$ Chatgpt and copilot are totally stumped. One thing to recognize is ...
4 votes
1 answer
284 views
Perfect powers as products of two Fibonacci or Lucas numbers
The well-known Fibonacci sequence $(F_n)_{n\ge0}$ and Lucas sequence $(L_n)_{n\ge0}$ are defined by $$F_0=0,\ F_1=1,\ \text{and}\ F_{n+1}=F_n+F_{n-1}\ \text{for}\ n=1,2,3,\ldots,$$ and $$L_0=2,\ L_1=1,...
- 17.6k
18 votes
2 answers
916 views
The golden ratio equation in terms of functions $f:\mathbb{N}\to\mathbb{N}$
The golden ratio $\varphi$ is the solution of the golden ratio equation $$\varphi^2 = \varphi + 1.$$ This led me to wonder whether such an equation can have a solution in the realm $\mathbb{N}^\mathbb{...
- 53.5k
0 votes
1 answer
161 views
On a certain Fibonacci identity
I'm examining the expression involving the Fibonacci numbers $$-231F_{2n+1}^3+264F_{2n+1}^2F_{2n}+198F_{2n+1}F_{2n}^2-33F_{2n}^3-308F_{2n+1}^2+308F_{2n+1}F_{2n}+308F_{2n}^2+231F_{2n+1}-33F_{2n}+321.$$ ...
- 1,205
6 votes
1 answer
432 views
Lucas number multiples of Fibonacci pairs
$\newcommand{\GCD}{\operatorname{GCD}}$ For $n=0,1,2,\ldots,$ let $F_n=0,1,1,2,3,5,\ldots$ and $L_n=2,1,3,4,7,11,\ldots$ be the Fibonacci and Lucas sequences. I expect the following is well known, but ...
- 301
2 votes
1 answer
205 views
$R$-recursion for Fibonacci numbers using signed Catalan numbers
Let $F_n$ be A000045 (i.e., Fibonacci numbers). Here $$ F_n = F_{n-1} + F_{n-2}, \\ F_0 = 0, F_1 = 1. $$ Let $C_n$ be A000108 (i.e., Catalan numbers). Here $$ C_n = \frac{1}{n+1}\binom{2n}{n}. $$ Let $...
user231922
0 votes
1 answer
325 views
Divide angles by coefficients relate to Fibonacci sequence
In the left Figure, consider a right triangle $OPA$ with $\angle {AOP} = 90^\circ$. Let $\ell$ be the reflection of $PO$ in $PA$ and $\ell$ meets $OA$ at $A_1$. Let $O_1$ be the center of the circle $(...
- 2,198
1 vote
1 answer
113 views
Sequence derived from transform of a given vector (with Fibonacci as partial sums)
Let F_n be A000045 (i.e., Fibonacci numbers). Here $$ F_n = F_{n-1} + F_{n-2}, \\ F_0 = 0, F_1 = 1 $$ Let $\operatorname{wt}(n)$ be A000120 (i.e., number of ones in the binary expansion of $n$). ...
user231922
2 votes
2 answers
282 views
Negated Fibonacci and the floor function
Let $F_n$ be A000045 (i.e., Fibonacci numbers). Here $$ F_n = F_{n-1} + F_{n-2}, \\ F_0 = 0, F_1 = 1, \\ F_{-n} = (-1)^{n-1}F_n $$ I conjecture that $$ F_{-n} = \left\lfloor\frac{n+1}{2}\right\rfloor ...
user231922
16 votes
1 answer
710 views
Limit involving the fractional part and the Fibonacci numbers
Helo, Let $F(n)$ be the $n$th Fibonacci number, if $\left\{ x\right\}$ denotes the fractional part of $x$, how proving $$\lim_{n\rightarrow\infty}\frac{1}{2n}\sum_{k=1}^{2n}\left\{ \frac{F(2n)}{F(k)}\...
- 693
3 votes
1 answer
264 views
Proof of an unknown source Fibonacci identity with double modulo
Many years ago, I saw the following Fibonacci identity from somewhere online, without proof: Let usual $F(n)$ be Fibonacci numbers with $F(0) = 0, F(1) = 1$, then we have $$F(n) = \left(p ^ {n + 1} \...
- 131
2 votes
0 answers
106 views
Splitting natural numbers into subsets with sums equal to A066258
Let $F(n)$ be A000045 i.e. Fibonacci numbers. Here $$ F(n) = F(n-1) + F(n-2), \\ F(0) = 0, F(1) = 1 $$ Let $a(n)$ be A066258 i.e. $$ a(n) = F(n)^2F(n+1) $$ Let $b(n)$ be A345253 i.e. maximal ...
user231922
20 votes
2 answers
824 views
A rational function related to Fibonacci numbers
Let $F_n$ denote a Fibonacci number ($F_1=F_2=1$, $F_{n+1}=F_n+F_{n-1}$ for $n\geq 2$). Define $$\prod_{k=1}^n (1+x^{F_{k+1}}) = \sum_j f(n,j)x^j. $$ For a positive integer $r$ let $$ v_r(n) = \sum_j ...
- 53.6k
4 votes
0 answers
216 views
The smallest sequence without differences among Fibonacci numbers
Given a subset $\mathcal S\subset \mathbb N\setminus\{0\}$ of (strictly) positive integers, we can consider subsets $A$ of $\mathbb N$ (or $\mathbb Z$) with no differences in $\mathcal S$. Examples: ...
- 18.4k
14 votes
3 answers
1k views
On the finite sum of reciprocal Fibonacci sequences
I want to check if $$\left\lfloor \left( \sum_{k=n}^{2n}{\frac{1}{F_{2k}}} \right)^{-1} \right\rfloor =F_{2n-1}~~(n\ge 3) \tag{$*$}$$ where $\lfloor x \rfloor$ is th floor function. The Fibonacci ...
- 147
3 votes
1 answer
205 views
Golden ratio base
Let $\phi$ be the golden ratio and look at real numbers as expansions in digits from base $\phi + 1$. Has this base been considered or studied anywhere? Note that integers in this base are palindromes ...
- 1,225
0 votes
1 answer
234 views
Density of "Fibonacci friends"
Let $F$ be the set of all integers $n>1$ such that in the Fibonacci sequence modulo $n$, the value $0$ occurs infinitely often. What is the value of $\lim\sup_{n\to\infty}\frac{|F\cap\{0,\ldots,n\}|...
- 53.5k
0 votes
1 answer
238 views
Fibonacci and product polynomials
The motivation for my current question arises from this MO post by R. Stanley. Caveat. There's a slight alteration. With the convention $F_1=F_2=1$ for the Fibonacci numbers, define the polynomials $...
- 43.7k
0 votes
0 answers
110 views
Avoiding the Fibonacci numbers
For given positive integers $a$ and $b$, let $(a,b)$ be "special" if $an+b$ is not a Fibonacci number for every positive integer $n$. For instance, $(8,4)$ and $(8,6)$ are special. There are ...
- 248
3 votes
0 answers
494 views
Conjecture about primes and Fibonacci numbers
I posted this conjecture on math.stackexchange, but I received no answer proving or disproving it: if $ m > 4 $ is a positive integer not divisible by $ 2 $ or $ 3 $, it's ever possible to find a ...
- 557
6 votes
0 answers
157 views
Equivalence of primes based on the partition of their Pisano periods
The period of Fibonacci numbers modulo $m$ is called Pisano period and its length is denoted as $\pi(m)$. Define the Pisano partition of $m$ as the set partition of the indices $\{0,1,\dotsc,\pi(m)-1\}...
- 38.3k
3 votes
0 answers
168 views
Reference for formula expressing products of two Fibonacci numbers in Zeckendorf-basis
It is well-known folklore that every natural integer has a unique Zeckendorf expansion as a sum over a finite set of Fibonacci numbers containing no pair of consecutive Fibonacci numbers. It is easy ...
- 18.4k
1 vote
1 answer
258 views
Explicit formula for Fibonacci numbers; compositions of $n$
A Fibonacci-type sequence is a sequence with two seed-values, $F_1$ and $F_2$, and which, for all $n>2$, abides by the recurrence relation $F_n = F_{n-1} + F_{n-2}$. If $F_1 = F_2 = s$, then the $n$...
- 13
16 votes
2 answers
623 views
Number of coefficients equal to $k$ in certain "Fibonacci polynomials"
Let $F_i$ denote the $i$th Fibonacci number (with $F_1=F_2=1$). Define $$ P_n(x) = \prod_{i=1}^n (1+x^{F_{i+1}}). $$ Let $\nu_k(n)$ denote the number of coefficients of the polynomial $P_n(x)$ that ...
- 53.6k
3 votes
1 answer
1k views
Why doesn't the number of ones in the binary representation of Fibonacci numbers grow linearly? [closed]
I am a third-year computer science student. I am interested, why doesn't the number of ones in the binary representation of Fibonacci numbers grow linearly? I would expect it to grow linearly all the ...
- 221
4 votes
0 answers
182 views
Binary iterations, Fibonacci numbers and permutation of natural numbers
Let $\operatorname{wt}(n)$ be A000120, i.e. the number of $1$'s in binary expansion of $n$ (or the binary weight of $n$). Also let's consider $$\ell(n)=\left\lfloor\log_{2} n\right\rfloor$$ and $$T(n,...
user231922
0 votes
0 answers
253 views
What are the hidden assumptions behind Harvey Friedman's claim, CSR?
I'm doing some archeology and trying to understand a claim. As summed up by David Roberts, on the FOM list in 2011: Let the statement "every infinite sequence of rationals in [0,1] has an ...
- 446
1 vote
1 answer
377 views
Tiling a square with similar non-congruent rectangles. What is the aspect ratio of the rectangles as n grows large?
I recently saw a question here on mathoverflow: «For what n and t can a square be partitioned into n similar rectangles in t congruence classes?», where Joseph Gordon gave a proof that, indeed, a ...
- 13
4 votes
1 answer
244 views
Why do convoluted convolved Fibonacci numbers pop up from this triangle?
Start with this triangle (OEIS A118981). This triangle is simple to generate with the following recurrence relation (though $T(0,0)$ ends up different from the OEIS version): $$ T(0,0) = 2;T(1,0) = 1;...
- 194
2 votes
0 answers
388 views
My Fibonacci Formula (with combinatorics) [closed]
I'm a high school student, and was playing around with pascals triangle. and ended up taking (weird) diagonals. And I saw Fibonacci numbers, from the sum of the diagonals. Pascall's triangle is just ...
- 21
20 votes
4 answers
2k views
Non-enumerative proof that, in average, less than 50% of tiles in domino tiling of 2-by-n rectangle are vertical?
Is there a non-enumerative proof that, in average, less than 50% of tiles in domino tiling of 2-by-n rectangle are vertical? It is a nice exercise with rational generating functions (or equivalently, ...
- 25.9k
8 votes
1 answer
442 views
Possible small mistake in Bilu-Hanrot-Voutier paper on primitive divisors of Lehmer sequences (?)
I think that I might have spotted I small mistake (a missing $5$-defective Lehmer pair) in the classification of terms of Lehmer sequences without primitive divisors given in: 1 Bilu, Hanrot, and ...
- 75
3 votes
0 answers
297 views
Is there a closed form of $ \displaystyle \sum_{k=0}^{\infty}{\frac{\phi^{xk}}{k!_F}}$
where $\phi = \frac{1+\sqrt{5}}{2}$ and $k!_F$ is the fibonorial of $k$, or the product of the first $k$ Fibonacci numbers? My hunch is that, this can be represented as a function in terms of the ...
2 votes
1 answer
216 views
Is this case of a generalised partition equivalent to Fibonacci numbers?
Let $k=m+\sum^{m+1}_{j=1} a_j$ such that $a,m,k\in\mathbb{N}$ and $a_1$ or $a_{m+1}\geq 0$ with all other $a\geq1$. Note that we assume natural numbers start from $0$ and we have the restriction that $...
- 79
-1 votes
1 answer
233 views
A generalization of Vajda's identity [closed]
I discovered the identity below which generalizes Vajda's identity concerning Fibonacci Numbers. The identity states that: if $F_r$ is the rth Fibonacci number, then $$F_{n+i+x-z}F_{n+j+y+z}-F_{n+x+y-...
- 123
0 votes
0 answers
126 views
Requesting proof of closed form of sum involving Fibonacci and Lucas numbers
$$ \sum_{n=0}^{k+1}\frac{3F_{n+1}-L_{n+1}}{2n!}\frac{(k+1)!}{(k-n+1)!}x^{k-n+1}=(\varphi+x)^k\left(\frac{\sqrt{5}}{5}-\frac{\sqrt{5}-5}{10}x\right)+(\psi+x)^k\left(\frac{\sqrt{5}+5}{10}x-\frac{\sqrt{5}...
12 votes
2 answers
1k views
The Fibonacci sequence modulo $5^n$
Let $(F_k)_{k=0}^\infty$ be the classical Fibonacci sequence, defined by the recursive formula $F_{k+1}=F_k+F_{k-1}$ where $F_0=0$ and $F_1=1$. For every $n\in\mathbb N$ let $\pi(n)$ be the smallest ...
- 44.2k
4 votes
1 answer
334 views
Fibonacci with seeds, modulo $n$
Let $n\in\mathbb{N}$ be an integer with $n>1$. For $x_0, x_1 \in \mathbb{Z}/n\mathbb{Z}$ we define the map $\text{fib}_{n, x_0, x_1}: \mathbb{N} \to \mathbb{Z}/n\mathbb{Z}$ by $0 \mapsto x_0, 1 \...
- 53.5k
2 votes
0 answers
137 views
Can all (inverse) trigonometric functions with periodic iterates be characterized?
I wonder whether all (composites of) trigonometric and inverse trigonometric functions with periodic functional iterations can be found. In order to specify what I mean by that, let's introduce some ...
- 5,025
7 votes
1 answer
330 views
On nontotient Fibonacci numbers
This question is related to sequence of numbers $t$ such that $F_{6t}$ is a nontotient where $F_n$ represents the sequence of Fibonacci numbers for $n\geq 0$. The online encyclopedia Wikipedia has the ...
- 701
18 votes
1 answer
697 views
Complexity of a Fibonacci numbers discrete log variation
In my work I encountered the following FIBMOD PROBLEM: Given $k,m$ in binary, decide if there exists $n$ such that $\, F_n = k \,$ (mod $m$). Here $F_n$ is a Fibonacci number. This is a variation ...
- 17.4k
47 votes
5 answers
4k views
Fibonacci series captures Euler $e=2.718\dots$
The Fibonacci recurrence $F_n=F_{n-1}+F_{n-2}$ allows values for all indices $n\in\mathbb{Z}$. There is an almost endless list of properties of these numbers in all sorts of ways. The below question ...
- 43.7k
20 votes
3 answers
4k views
Reciprocals of Fibonacci numbers
Is the sum of the reciprocals of Fibonacci numbers a transcendental?
- 413
9 votes
2 answers
1k views
Elementary problem with Fibonacci numbers
I need help in proving one elementary result with Fibonacci numbers. Prove that for $n>2$, the product $F_1 \cdot F_2 \cdots F_n$ cannot be a perfect square, where $F_1 = F_2 = 1, F_{n+1}=F_n + F_{...
- 189