Questions tagged [golden-ratio]
Questions relating to the golden ratio $\varphi = \frac{1+\sqrt{5}}{2}$
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Algorithm to find floors of multiples of the golden ratio
What is an algorithm to calculate $\lfloor n\phi \rfloor$ given some integer $n$, where $\phi$ is the golden ratio?
I am thinking the easiest way will involve calculating multiples of its continued ...
0
votes
1answer
13 views
Which relation symbol goes between $F_m$ and $\phi^m$?
I have already asked a similar question where the tilde notation was used (and context can be taken from there). Now I think that tilde is not the correct symbol to go between these two functions ...
0
votes
1answer
19 views
ring of integers with golden ratio
How to show that the ring of integers of $\mathbb{Q}(\sqrt{5})$, i.e., $\mathbb{Z}[\phi]$ is a principal ideal domain (where $\phi$ is the golden ratio)?
I want to prove that using as elementary ...
4
votes
2answers
67 views
$a,b \in \mathbb Z$ we have that $a+ b\phi \in \mathbb Z[\phi]^* \iff a^2 +ab-b^2 = \pm 1$
I am thinking about the following question:
Let $\phi= \frac{1+ \sqrt 5}{2}$ be the golden ratio, and define:
$$ \mathbb Z [\phi]=\{a+ b\phi : \quad a, b \in \mathbb Z\} $$
We wish to prove ...
4
votes
1answer
102 views
Numberphile's “plastic number” video, question regarding “calipers” used
Sorry for the poor title, but this question is awkward.
Numberphile has a new video about the plastic number, and it demonstrates it with a set of calipers with 4 prongs.
https://youtu.be/...
4
votes
1answer
114 views
Fibonacci elegance sought for $F_{f (n)} + F_{f(n)-1}$
Updated on Friday 15th March 2019 at 5 pm in the light of comments received over the last 24 hours.
The original question was; given the well known variation of Binet's Formula:
$$F_n = \frac{\phi^n ...
1
vote
1answer
51 views
Remembering the golden ratio
I always forget how one can deduce the golden Ratio and its property. I hope somebody can explain me the chain of thoughts of its introduction in the book.
By property I mean that the reciprocal of ...
0
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2answers
54 views
On the golden ratio and odd perfect numbers
Here is my question:
Is $I(n^2) - 1 > 1/I(n^2)$ true when $I(n^2)=\sigma(n^2)/n^2$ is the abundancy index of $n^2$ and $q^k n^2$ is an odd perfect number with special prime $q$ satisfying $k>...
1
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0answers
87 views
A Golden Ratio Functional Equation Sequence
I was looking at the equation $f^{-1}(x)=\int f(x)dx$ recently.
One can note that it has an easy real-valued solution $f(x)=\phi^{\frac{\phi-1}{\phi}}x^{\phi-1}$ (by guessing for a solution of the ...
2
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0answers
174 views
Is there a meaningful link between the golden ratio and chaos theory?
I heard it casually mentioned by strangers but am unable to find any information about this.
Is there a meaningful link between the golden ratio and chaos theory?
The closest thing I could find is ...
0
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0answers
15 views
How do you construct a golden spiral with recursive rotating and scaling segments?
I don't have a function generator at hand, so instead, I'll use segments.
For constructing a golden spiral, I know that a good start is to take a segment of 1 unit, rotate it 90 degrees, then scale ...
4
votes
2answers
149 views
Studying on this sum interesting $\sum_{n=1}^{\infty}\frac{{2n \choose n }}{4^n n}$
I was studying this particular sum $$\sum_{n=1}^{\infty}\frac{{2n \choose n}}{4^n n}$$ and eventually I ended up with is sum $(1)$
$$\sum_{n=1}^{\infty}\frac{{2n \choose n}}{(-4\phi)^n}\cdot\frac{1}{(...
-3
votes
1answer
102 views
Harmonic number and the golden ratio
I just don't understand how $(1)$ can have this simple closed form.
$$\sum_{k=1}^{\infty}\frac{{2k \choose k}}{(-16)^k}[H_k-H_{k+1}]=\phi^{-6}\tag1$$
Where $\phi=\frac{1+\sqrt{5}}{2}$, is the ...
29
votes
1answer
850 views
What is the maximum value of this nested radical?
I was experimenting on Desmos (as usual), in particular infinite recursions and series. Here is one that was of interest:
What is the maximum value of $$F_\infty=\sqrt{\frac{x}{x+\sqrt{\frac{x^2}{x-...
10
votes
3answers
821 views
A Series For the Golden Ratio
Question: Can we show that $$\phi=\frac{1}{2}+\frac{11}{2}\sum_{n=0}^\infty\frac{(2n)!}{5^{3n+1}(n!)^2} $$; where $\phi={1+\sqrt{5} \above 1.5pt 2}$ is the golden ratio ?
Some background and ...
1
vote
2answers
56 views
Why is $F_n = r^n$ a solution of the difference equation if $r$ satisfies $r^2-r-1=0$?
The following is from p.4 of https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/ch3.pdf
The terms in the Fibonacci sequence are uniquely determined by the linear
difference equation
$$...
2
votes
3answers
210 views
How to show that $\sum_{n=1}^{\infty}\frac{\phi^{2n}}{n^2{2n \choose n}}=\frac{9}{50}\pi^2$
Given:$$\sum_{n=1}^{\infty}\frac{\phi^{2n}}{n^2{2n \choose n}}=\frac{9}{50}\pi^2$$
Where $\phi=\frac{\sqrt{5}+1}{2}$
How can I we show that the above sum is correct?
I have checked numerically, it ...
4
votes
0answers
47 views
Functional Equation involving Golden Ratio [duplicate]
Find all $h:\mathbb{N}\rightarrow \mathbb{N}$ such that
$$h(h(n)) + h(n+1) = n+2$$
I tried this, but wasn't able to make any progress after a while. So, in vain,
I looked at the solution. The ...
0
votes
1answer
103 views
Continued fraction of $\phi^3$
I found that $$\phi^3=4+\cfrac1{\small{4+\cfrac1{4+\cfrac1{4+\cfrac1{4+\ddots}}}}}$$
How should I prove this?
Attempt:
Suppose$$x= 4+\cfrac1{\small{4+\cfrac1{4+\cfrac1{4+\cfrac1{4+\ddots}}}}}$$
To ...
0
votes
1answer
57 views
Continued fractions approximation using golden ratio
Hello today my friend helped me with my problem, but he did not give me any additional informations why it works like that.
Let's suppose that I need to get ln(n) using continued fractions. He told ...
5
votes
4answers
126 views
Does $a_{n}/a_{n-1}$ converge to the golden ratio for all Fibonacci-like sequences?
Yesterday a friend challenged me to prove that
$$\lim_{n\rightarrow\infty}\frac{a_n}{a_{n-1}}=\varphi\; ,$$
where $\varphi$ is the golden ratio, for the Fibonacci series.
I started rewriting the ...
0
votes
1answer
42 views
Proofs regarding the golden number
Given a succession $r(n)= 1 +\frac{1}{r(n-1)}$
where $r (1)=1$
and
golden number $\phi =\frac{1+\sqrt{5}}{2}$.
How do I prove that
$$\left\lvert r(n)-\phi\right\rvert \leq \frac{1}{\phi ...
0
votes
0answers
28 views
Express solutions in terms of $\phi$
Previously, I solved the special transcendental equation $x=e^{t/\ln(x)}$. The solution is: $x=e^{-\sqrt{t}}$ for $0<x<1.$
One can define an equation: $e^{s/\ln(x)}=e^{t/\ln(1-x)},$ for $s,t \...
0
votes
1answer
59 views
More irrational than the Golden Ratio?
According to this video, $\varphi$ is the most irrational number, due to its continued fraction form having $1$, the smallest natural number, in the denominators.
Is it not possible to construct a "...
13
votes
4answers
308 views
What is this function related with continued fractions?
Playing with continued fractions, I came with the idea of iterating the limit of the simplest one:
$$1 + \cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cdots}}}}}\ = \Phi$$
And then I ...
5
votes
1answer
82 views
Ratio between farthest and second farthest distance
$n\geq 3$ points lie in three-dimensional space. What is the largest $c(n)$ such that there always exists a point for which the ratio between the distance to the farthest point from it and the ...
1
vote
2answers
72 views
Computing a Geometrical ratio $\frac{a}{b}$.
$XYZ$ is an equilateral triangle as shown on the image below.
The aim is to find the ratio $\frac{a}{b}$.
So far from the picture, it is easy to see that $b= \frac{YZ}{2}$.
Does anyone have an ...
18
votes
1answer
297 views
Continued fraction involving Fibonacci sequence
What is the limit of the continued fraction:
$$\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{5+\cfrac{1}{8+\cdots}}}}}}\ $$
that involves the Fibonacci sequence terms as denominators? I'...
0
votes
1answer
40 views
Ratio of 0's to 1's in the Fibonacci Word is the golden ratio
Define $S_0=0, S_1=01$. Then for $n\geq 2$ we define $S_n=S_{n-1}S_{n-2}$ (concatenating the previous sequence and the one before that). We obtain a limiting sequence, which we call the inifinite ...
0
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0answers
50 views
Is there a “natural” entire series associated to the Riemann zeta function whose radius of convergence is $\frac{1}{\sqrt{5}}$?
As a follow-up to Is there a hidden connection between RH and the golden ratio?, let's consider the plane $ P $ whose intersection with the Riemann sphere is the circle I denoted by $ \Gamma_{\Delta}...
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0answers
43 views
How does this drawing of a man's body in a pentagram suggest relationships to the golden ratio?
According to Wikipedia it does, but I can't see how.
Wikipedia context: https://en.wikipedia.org/wiki/Golden_ratio#/media/File:Pentagram_and_human_body_(Agrippa).jpg
13
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3answers
728 views
A pattern appearing in the powers of $\phi$
\begin{align}
\phi^5 &= 11,\underline{0}901699\cdots\\
\phi^6 &= 17,\underline{9}44271\cdots\\
\phi^7 &= 29,\underline{6}34441\cdots\\
\phi^8 &= 46,\underline{9}7871\cdots\\
\phi^9 &...
0
votes
1answer
73 views
Show that $\int_0^{\infty} (\exp(x) - 2^{\sqrt 5 +1} + 1) (\coth(x) - 1) x^{\sqrt 5} dx = 0 $ [closed]
Show that
$$ \int_0^{\infty} (\exp(x) - 2^{\sqrt 5 +1} + 1) (\coth(x) - 1) x^{\sqrt 5} dx = 0 $$
I wonder how many distinct methods there are.
1
vote
1answer
165 views
Prime number and golden ratio
I think the following is true, but can't show it.
Let $p$ be a prime number, and let $f(x)$ be a polynomial which satisfies
$${\Bigl(1-x+\frac{1}{x}\Bigr)}^p-1=f(x)+f\Bigl(-\frac{1}{x}\Bigr).$$
Then
$...
0
votes
1answer
74 views
Golden and Silver ratios, how about the roots of $x^2+x-1=0$?
The golden and silver ratios are the roots of the equation $x^2-x-1=0$:
$$\frac{1\pm\sqrt{5}}{2}.$$
They show up in the formula of Fibonacci numbers:
$$F_n=\frac{1}{\sqrt{5}}\left(\frac{1+\sqrt{5}}{2}...
1
vote
3answers
75 views
Finding the exact value of $\sin 30°$ using golden ratio [closed]
$\sin \left(k\cdot 30^\circ\right) = \frac{\sqrt{2}}{4}\;\sqrt{\;4\;\pm_1\;\sqrt{\phi\,(a\phi+b\overline{\phi})}\;\pm_2\;\sqrt{\overline{\phi}\,(c\phi+d\overline{\phi})}\;}$
I'm currently trying to ...
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0answers
227 views
Golden Ratio in trigonometry
Assume that we have been asked to find the value of $\sin (18^\circ)$. We know that there are many ways to find it out. However, I'll be going with golden ratio! Let's draw a triangle whose apical ...
1
vote
1answer
109 views
Growth factor of the golden spiral
Wikipedia says, that
a golden spiral is a logarithmic spiral whose growth factor is φ, the golden ratio.
However, other sources[only in Czech, sorry] say that the "growth factor" of a spiral is ...
2
votes
1answer
50 views
How many ways can dominoes cover a $2 \times n$ rectangle? Justify proposed solution.
I was able to get that $d_n = d_{n-1}+d_{n-2}$
It isn't finished, because I have to solve this recursive equation.
I read about Binet's formula, but I don't know the steps between this
$$d_n = d_{n-...
6
votes
1answer
77 views
(Non-) Convergence of $\frac{1}{n} \sum_{k=0}^{n - 1} \exp\left(2i \pi [\frac{3 + \sqrt{5}}{2}]^k\right)$ when $n \to +\infty$
Let be $$\forall n > 0, S_n = \dfrac{1}{n} \sum\limits_{k=0}^{n - 1} \exp(2i\pi u_k),\quad \forall k \geq 0, u_k = \left(\dfrac{3 + \sqrt{5}}{2}\right)^k$$
I would like to prove or disprove the ...
6
votes
3answers
92 views
Matrix satisfying $A-I = A^{-1}$
Recall the (infinitely) continued fraction definition of the golden ratio
\begin{align}
\phi = 1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\cdots}}}}}
\end{align}
This is equivalent to ...
0
votes
1answer
91 views
Why is $x$ always a perfect square?
In the following equation: $$5 F_n^2 \pm 4 = x,$$
where $F_n$ is a Fibonacci number, and the $\pm 4$ shall be treated as $+4$ for even $n$ and $-4$ for odd $n$.
Now, if the above requirements are ...
0
votes
1answer
180 views
Fibonacci numbers and perfect squares
Can this be simplified any further?:
$\phi^{2n}$ + $\psi^{2n}$$-2(-1)^n$ $\pm4$
Where $\phi = (1+\sqrt5)/2 $
Where $\psi = (1-\sqrt5)/2 $
When n=even number, use +4
Whenn n=odd number, use -4
...
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votes
2answers
104 views
Powers of the golden ratio [closed]
Let $\phi$ be the golden ratio. I'm tasked to prove by other means than induction that $x$ in the next equation $$\phi^n =\phi F_n +x,$$ is actually a Fibonacci number.
I have tried to apply Binet's ...
1
vote
0answers
64 views
Prove that $\operatorname{fib}(n)$ is the closest integer to $\frac{\phi^{n}}{\sqrt{5}}$
From SICP:
Exercise 1.13: Prove that $\operatorname{fib}(n)$ is the closest integer to
$$\frac{\phi^{n}}{\sqrt{5}}, \text{ where } \phi=\frac{1+\sqrt{5}}2.$$ Hint: Let
$$\varphi=\frac{1-\sqrt{5}...
6
votes
1answer
102 views
On $3+\sqrt{11+\sqrt{11+\sqrt{11+\sqrt{11+\dots}}}}=\phi^4$ and friends
Let $\phi$ be the golden ratio. We know it has a beautiful infinite nested radical,
$$\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\dots}}}}=\phi$$
However, it is also the case that,
$$3+\sqrt{11+\sqrt{11+\sqrt{...
1
vote
3answers
47 views
Golden mean in this equation
Years ago I had started with this equation:
$2^{1/3}=(R/2)(\sqrt{1+8/R^3}-1)$
And arrived at the result
$2^{1/3}=\phi R$
Where phi, the golden ratio, is (sqrt(5) +1)/2.
But at the moment can't ...
2
votes
0answers
132 views
$\int\limits_{0}^{t}(\exp{(\sqrt{x}-x^2)} )^ \text{erf(x)}\ \text{d}x,$ VS golden ratio For $t\geq 2$ ,
I have tried to get aother integral representation for the Golden ratio i have got the following representation . This integral is defined as : $$I(t)=\int\limits_{0}^{t}(\exp{(\sqrt{x}-x^2)} )^ \text{...
14
votes
1answer
316 views
Golden Rectangle into Golden Rectangles
Can these golden rectangles be rearranged to exactly cover the underlying cyan golden rectangle?
That's the entire question. All that follows is related discussion. I want to make a more elegant ...
3
votes
1answer
100 views
Are $\varphi$ and $\varphi^3$ the only powers of $\varphi$ that are also metallic ratios?
Given the sequence generator function: $$F_\lambda(n+2)=\lambda F_\lambda(n+1)+F_\lambda(n);\quad F_\lambda(0)=0, F_\lambda(1)=1$$ where $\lambda=1$ corresponds to the Fibonacci sequence, $\lambda=2$ ...
