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

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Find $x$ as the given $n$th term in the Fibonacci sequence?

With a given $n$ and I am trying to find the value of $x$, as in: $$Fib(x)=n$$ Using the formula for Fibonacci sequence, where $\varphi$ is the Golden Ration ($\approx1.61803399\ldots$) $$Fib(z) = ...
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2answers
504 views

Why does this graph intercept both axes at the golden ratio?

Earlier, I was playing around with the Desmos Graphing Calculator, and I discovered that the following formula intercepts both the x and y axes at the golden ratio. I know that it makes sense, but I ...
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0answers
25 views

Why does the tribonacci constant have a trilogarithm ladder?

When I came across the dilogarithm ladders of Coxeter and Landen, namely, $$\text{Li}_2(\alpha^6)= 4\text{Li}_2(\alpha^3)+3\text{Li}_2(\alpha^2)-6\text{Li}_2(\alpha)+\tfrac{7}{5}\zeta(2)\tag1$$ ...
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How is the Binet's formula for Fibonacci reversed in order to find the index for a given Fibonacci number?

a question about the Fibonacci sequence: $$F_n =\frac{\phi^n-(-\frac{1}{\phi})^n}{\sqrt{5}}$$ This is the Binet's formula for the nth Fibonacci number. if I reverse it I can get: ...
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1answer
94 views

Relationship between Pi and Phi using the Great Pyramid of Giza?

In a documentation about the Great Pyramid of Giza, I heared following three theses about its measurements and the numbers $\pi$ and $\phi$ (the golden ratio). Measurement The Great Pyramid of ...
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58 views

Mean and Variance of Fibonacci Numbers

I would like to ask the community for feedback regarding the following two conjectures of mine: $\textbf{Conjecture 1}$ Let $\mathcal{F}_N^- = \{F_n:-N \leq n < 0\}$, i.e. be the set of Fibonacci ...
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1answer
86 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+ ...
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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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29 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 ...
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63 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 ...
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29 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 + ...
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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 ...
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1answer
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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 ...
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3answers
163 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 ...
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2answers
58 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 ...
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3answers
267 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 ...
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1answer
106 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!
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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 ...
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2answers
87 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 ...
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1answer
102 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.
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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 ...
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3answers
153 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 ...
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$\pi$, $e$, $\phi$, 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 ...
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80 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 $\phi$ ...
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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 ...
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166 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. ...
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1answer
67 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 ...
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1answer
69 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 ...
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1answer
77 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 ...
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39 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 ...
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5answers
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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 ...
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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 ...
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3answers
321 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 ...
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4answers
709 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 $$ ...
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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 ...
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5answers
174 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 ...
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105 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 ...
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530 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 ...
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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}\\ ...
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914 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 ...
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1answer
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$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 ...
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1answer
68 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: ...
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1answer
169 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 ...
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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)\\ ...
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704 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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137 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) ...
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3answers
768 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 ...
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3answers
165 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
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108 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} = ...
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1answer
184 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 ...