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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 ...
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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 ...
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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
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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
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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
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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 ...
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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 ...
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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>...
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0answers
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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 ...
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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 ...
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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 ...
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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}{(...
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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 ...
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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-...
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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 ...
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2answers
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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 $$...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 \...
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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 "...
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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
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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 ...
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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 ...
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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'...
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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 ...
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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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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
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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 &...
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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.
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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 $...
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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}...
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3answers
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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 ...
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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 ...
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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-...
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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 ...
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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 ...
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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 ...
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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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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 ...
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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}...
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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{...
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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 ...
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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{...
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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 ...
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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, $\lambda=2$ ...