Tagged Questions

Questions relating to the golden ratio $\varphi = \frac{1+\sqrt{5}}{2}$

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1
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1answer
66 views

Continued fraction of the golden ratio

It is known, that the continued fraction of $\phi = \frac{1+\sqrt{5}}{2}$ is $[\bar{1}]$. This can be shown via the equation $x^2-x-1=0$: $$ x^2-x-1=0 \Rightarrow x = 1+\frac{1}{x} = 1+ ...
0
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0answers
25 views

Golden Ratio Sandbox

This might be a little long so please bear with me. The Golden Ratio $\phi$ is defined as the single positive root of the polynomial $p(t) = t^2 - t - 1$. One can think of it as a line divided into ...
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0answers
17 views

Relationship of powers of Phi to Lucas Numbers

I was watching a Numberphile and the interviewee was explaining various attributes of Lucas Numbers and he made the statement about creating a sequence by starting with the Golden Ratio and raising it ...
2
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0answers
55 views

Arctangents, Fibonacci numbers, and the golden ratio

In the course of doing scratchwork to answer this question, I had occasion to write the trigonometric identity $$ \arctan x- \arctan(1-x) = \arctan\left( \frac{1-2x}{x^2-x-1} \right). $$ Now notice ...
0
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1answer
24 views

Sums of Golden Ratio Powers

I had a question regarding the following sum. Let $\phi$ be the golden ratio and $N$ be an even integer. \begin{array}{lcl} \sum_{n=1}^N (-\phi)^n & = & -\phi + (-\phi)^2 + (-\phi)^3 + ...
5
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0answers
65 views

Legitimate papers refuting the significance of the golden ratio in art?

I'm not sure this is the right place to ask about this, but is there any legitimate peer-reviewed paper refuting the significance of the golden ratio in art? I can find numerous websites and blogs ...
1
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1answer
73 views

I don't know how to solve equations used in the golden ratio

Today i was reading something from golden ratio and i don't understand how some equations where solved for example: Im told that $\phi_{n+1}=B_{n+1} + \frac {A_n}{B_n}$. What I don't understand is ...
3
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3answers
157 views

Proving that $\frac{\phi^{400}+1}{\phi^{200}}$ is an integer.

How do we prove that $\dfrac{\phi^{400}+1}{\phi^{200}}$ is an integer, where $\phi$ is the golden ratio? This appeared in an answer to a question I asked previously, but I do not see how to prove ...
4
votes
2answers
54 views

Is there anything special about a graph with the golden ratio in its spectrum?

Given a simple connected graph $g$ with adjacency matrix $\mathbf{A}$. Let the spectrum $\lambda_1 < \lambda_2 < \ldots < \lambda_N$ be the eigenvalues of the equation $\mathbf{A} v=\lambda ...
2
votes
3answers
258 views

Simplifying the sum of powers of the golden ratio

I seem to have forgotten some fundamental algebra. I know that: $(\frac{1+\sqrt{5}}{2})^{k-2} + (\frac{1+\sqrt{5}}{2})^{k-1} = (\frac{1+\sqrt{5}}{2})^{k}$ But I don't remember how to show it ...
3
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1answer
95 views

Golden Ratio of Primes (Amateur)

Unable to find information elsewhere, so I'll try here. What two consecutive primes are closest to producing the Golden Ratio? Or two of any Primes? Has this been determined? Thanks!
7
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2answers
182 views

Does $\sum\limits_{n=1}^{\infty}\frac{1}{P_n\ln(P_n)}$ converge to the golden ratio?

The sum $\displaystyle\sum\limits_{n=2}^{\infty}\frac{1}{n\ln(n)}$ does not converge. But the sum $\displaystyle\sum\limits_{n=1}^{\infty}\frac{1}{P_n\ln(P_n)}$ where $P_n$ denotes the $n$th prime ...
4
votes
2answers
84 views

Connection with golden ratio?

Consider the following problem: Let $p\in\mathbb{Z}[x]$ be a polynomial with integer coefficient. Suppose that the leading coefficient is 1, all roots are real and in $(0, 3)$. Find all ...
8
votes
1answer
100 views

Prove $_2F_1\!\left(\frac76,\frac12;\,\frac13;\,-\phi^2\right)=0$

Please help me to prove the identity $$_2F_1\!\left(\frac76,\frac12;\,\frac13;\,-\phi^2\right)=0,$$ where $\phi=\frac{1+\sqrt5}2$ is the golden ratio.
2
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0answers
61 views

Is there a golden pyramid?

Related to golden ratio: Golden rectangle is said to be the most aestheticaly pleasing among rectangles: This question mentions golden triangles: On the other hand, another question mentions ...
3
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3answers
136 views

$\int_{0}^{\pi/2}\ln\left(1+4\sin^4 x\right)\mathrm{d}x$ and the golden ratio

We already know that, for any real number $t$ such that $t\geq-1$, $$ \int_{0}^{\pi/2} \ln \left(1+t \sin^2 x\right) \mathrm{d}x = \pi \ln \left( \frac{1+\sqrt{1+t}}{2} \right). $$ Prove that ...
5
votes
2answers
135 views

$\pi$, $e$, $\phi$, and sunflowers

While reading some internet materials on design, I came across this picture and comment: I found it a little bit surprising. I knew that the real sunflower follows golden ratio in some way (but I ...
2
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0answers
79 views

Can I use the equality $\phi^2=\phi+1$ without proving it?

I am looking at the following exercise: $$\text{ Show with induction,that the } i^{th} \text{ number Fibonacci satisfies the equality: } $$ $$F_i=\frac{\phi^i-\hat{\phi}^i}{\sqrt{5}}$$ where $\phi$ ...
5
votes
2answers
1k views

What is golden ratio doing in this computer code?

In this file (related to random number generation), there is following line: private const int MSEED = 161803398; which reminds on golden ratio. How come ...
9
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3answers
165 views

Generalizations of $\sum_{m=3n+2}^{\infty}\phi^m=\phi^{3n}$ and $\sum_{m=13n+1}^{\infty}(\sqrt2-1)^m=\dfrac{(\sqrt2-1)^{13n}}{\sqrt2}$

I noticed that the following identies hold with the help of wolfram alpha and oeis. I'm sure they're well-known, but I'd like to know how they generalize. ...
1
vote
1answer
66 views

Proof that golden angle successively divides the largest gap by the golden ratio?

The golden angle divides the circumference of a circle by the golden ratio. "If radial spokes are placed successively into the circle, each spaced by a golden angle increment, then each additional ...
0
votes
1answer
60 views

Function relating Euler's constant and the golden ratio

Okay, I was messing around on Excel with some coefficients and I stumbled onto this. Not sure if it converges but it gets pretty damn close around the 1024th term mark. Was wondering if somebody could ...
3
votes
1answer
72 views

Rational aproximations of golden ratio

I read a blogpost that mentions that for golden ratio, the sets of best rational approximations of the first kind and the second kind are the same. Is this true? If so, why? Are there other numbers ...
0
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0answers
35 views

Golden Ratio method question

I am reading a link on the Golden Ratio Method from http://mathfaculty.fullerton.edu/mathews/n2003/GoldenRatioSearchMod.html The part where it says If $f(c) \leq f(d)$ and only one new ...
11
votes
5answers
421 views

What's the value of $n+\cfrac{n}{n+\cfrac{n}{n+\cfrac{n}{\vdots}}}$ for $n\in\mathbb{C}$?

Write $$\phi_n\stackrel{(1)}{=}n+\cfrac{n}{n+\cfrac{n}{\vdots}}$$ so that $\phi_n=n+\frac{n}{\phi_n},$ which gives $\phi_n=\frac{n\pm\sqrt{n^2+4n}}{2}.$ We know $\phi_1=\phi$, the Golden Ratio, so ...
4
votes
2answers
142 views

Generalizations of the golden and silver ratios, and their significance

$\Phi$, or the golden ratio, is basically $\frac{a+b}{a}=\frac{a}{b}$. The silver ratio corresponds to a similar idea of: $\frac{2a+b}{a}=\frac{a}{b}$. I've read on Wikipedia that both of these ratios ...
3
votes
3answers
179 views

Fibonacci Sequence, Golden Ratio

I've been asked to show that $x_n \rightarrow L$ as $n \rightarrow \infty$ where $x_n = F_{n+1}/F_{n}$ for $n \in \mathbb{Z}^+$, where $F_n$ denotes the $n^{th}$ Fibonacci number. I am supposed to use ...
6
votes
4answers
702 views

Fibonacci numbers and golden ratio

Let $\Phi$ be the golden ratio and $F_n$ be the usual Fibonacci numbers. How can I derive the following formula? $$ \Phi = \lim_{n\rightarrow \infty} \sqrt[n]{F_n} $$ I know the usual relation $$ ...
1
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0answers
82 views

A sequence that converges to the golden ratio

This is elementary, but I found it somewhat surprising. Define $$ a_n = \frac{1+ a_{n-2}}{\sqrt{1+a_{n-1}}} \;,$$ where $a_1$ and $a_2$ are constants. For example, here is a plot for $a_1=5$ and ...
1
vote
5answers
129 views

Limit of Ratio of Adjacent Fibonacci numbers $\to \phi$ [duplicate]

We define the $n^{th}$ Fibonacci number as $a_1 = a_2 = 1$ and $a_n = a_{n-1} + a_{n-2}$ for $n \geq 3$. Consider $$ \lim_{n \to \infty} \frac{a_{n+1}}{a_n}. $$ I wrote a script and found that this ...
0
votes
2answers
101 views

Prove that a Fibonacci number is greater than $ φ^n$

How can I prove the following: If $f_n$ is a number of the Fibonacci sequence and φ= $\frac{1+\sqrt{5}}2$, then $f_n > φ^n$ for every $n >2$? I have tried using induction but I can't ...
4
votes
2answers
440 views

Prove that the limit of two consecutive fibonacci numbers EXISTS. [duplicate]

Using the definition of Fibonacci numbers, $F_n=F_{n-1}+F_{n-2}$, I can prove that the limit of $\frac{F_{n+1}}{F_n}$ as $n\to\infty$ is $\phi$ if we assume that the limit exists. How can we prove ...
0
votes
2answers
67 views

How does this proof of Fibonacci work

\begin{eqnarray*} F_{i+1}&=&F_{i} + F_{i-1}\\ &=&\frac{\phi^i-\hat{\phi^i}}{\sqrt5}+\frac{\phi^{i-1}-\hat{\phi^{i-1}}}{\sqrt5}\\ ...
1
vote
2answers
709 views

How To Determine If A Large Number is Prime?

For a very large number n, how many divisibility tests are required to establish if its prime? I know this has something to do with the Golden Number, but I can't figure out what. I did try searching ...
0
votes
1answer
40 views

$e_{n+1} = K e_n e_{n-1} $ is $|e_{n+1}| = C|e_n|^{\varphi}$?

if $ e_{n+1} = K e_n e_{n-1} $ ($K$ is a constant, and $e_n$ is a serise), then, $ | e_{n+1} | = C|e_n|^\varphi$($C$ is constant) and $\varphi$ is golden ratio. Is this true? If yes, How can I show ...
2
votes
1answer
66 views

Source for relationship between $d$-ary Fibonacci numbers and generalized golden ratio?

I am not a mathematician (but a computer scientist) and stumbled across the following in the analysis of an algorithm (Berthold Vöcking: How Asymmetry Helps Load Balancing). The author gives Knuth: ...
2
votes
1answer
163 views

Representation of integers as powers of the golden ratio

How to prove that any integer $n$ can be represented in the form of $$n= \phi^{z_1}+\phi^{z_2}+\phi^{z_3}+...+\phi^{z_m}$$ For $z_1$, $z_2$... $z_m$ $\in$ $\mathbb Z $ and $\phi =\frac{ \sqrt ...
3
votes
0answers
88 views

Finding relatives of the series $\varphi =\frac{3}{2}+\sum_{k=0}^{\infty}(-1)^{k}\frac{(2k)!}{(k+1)!k!2^{4k+3}}$.

Consider $\varphi=\frac{1+\sqrt{5}}{2}$, the golden ratio. Bellow are series $(3)$ and $(6)$ that represent $\varphi$ $$ \begin{align*} \varphi &=\frac{1}{1}+\sum_{k=0}^{\infty}\cdots&(1)\\ ...
3
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7answers
642 views

Is there an identity that links $\pi$ and $\phi$ (the golden ratio)? [duplicate]

Is there some identity that shows a connection between $\pi$ and the golden ratio, $\phi$?
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3answers
131 views

Fibonacci Sequence or Golden Ratio?

Using the polar coordinate system, $r$ increases directly with $\theta$. In other words, $r=k\theta$. Which of the following shapes is constructed? A) Fibonacci Sequence B) Golden Ratio C) ...
2
votes
3answers
597 views

Limit of the ratio of consecutive Fibonacci numbers [duplicate]

I have read in a book that the limit of the ratio of consequent Fibonacci numbers is the golden ratio. However, it was just mentioned thus not justified. So, my question is how would you derive the ...
2
votes
3answers
146 views

Golden ratio / Fibonacci which branch of math?

Friends, The Golden ratio / Fibonacci sequence are studied under which branch of math? Can you recommend some good textbooks on the subject? Thanks
3
votes
3answers
107 views

Best way to discover the 'golden ratio'

Let's say you live in a world where nobody ever discovered the Golden ratio. What's the most intuitive way to discover this proportion? Wikipedia defined it this way: $$\phi = \frac{a+b}{a} = ...
1
vote
1answer
157 views

Golden spiral created using golden rectangles vs pentagram

I am trying to create a graphic that shows the golden spiral created using a pentagram and the golden triangles contained therein. I have drawn out the pentagram and golden triangles and the ...
22
votes
2answers
297 views

On the Paris constant and $\sqrt[k]{1+\sqrt[k]{1+\sqrt[k]{1+\sqrt[k]{1+\dots}}}}$?

In 1987, R. Paris proved that the nested radical expression for $\phi$, $$\phi=\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\dots}}}}$$ approaches $\phi$ at a constant rate. For example, defining $\phi_n$ as ...
0
votes
1answer
506 views

Induction on Fibonacci Sequence and the Golden Ratio

I'm trying to prove $f_n \leq \left(\frac{1 + \sqrt{5}}{2}\right)^{n-1}$ with induction, and I'm stuck in the induction step. Basis: n = 2 $f_{2} \leq \left(\frac{1 + \sqrt{5}}{2}\right)^{2-1} ...
0
votes
3answers
56 views

Is there a solution of $(x-1)/x=1$ or $(x+1)/x=1$?

Is there a solution of $(x-1)/x=1$ or $(x+1)/x=1$? Layman is trying to reversing golden ratio. Tell a real solution and complex solution please. Intuition says there should be a real value of ...
23
votes
2answers
434 views

A closed form of $\int_0^\infty\frac{\sqrt[\phi]{x}\ \arctan x}{\left(x^\phi+1\right)^2}dx$

Is it possible to evaluate the following integral in a closed form? $$\int_0^\infty\frac{\sqrt[\phi]{x}\ \arctan x}{\left(x^\phi+1\right)^2}dx,$$ where $\phi$ is the golden ratio: ...
3
votes
4answers
164 views

Hint to prove that $\phi^n + \phi'^n$ is an integer.

I was solving some induction exercises but I found this that I could not solve. Let $n \in \mathbb{N}$, prove that $\phi^n + \phi'^n$ is an integer where $\phi=\frac{1+\sqrt{5}}{2}$ and ...
0
votes
1answer
223 views

Internal polygon formed by drawing diagonals in a regular polygon

In an n-sided (n>4) regular polygon, label the vertices {0, 1, ..., n-1}. For each vertex i, draw a pair of diagonals: from i to (i+2) mod n and from i to (i-2) ...