# Questions tagged [golden-ratio]

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

330 questions
33 views

### 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 ...
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 ...
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 ...
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 ...
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/...
114 views

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 ...
850 views

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 ...
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 ...
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 ...
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 ...
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 ...
42 views

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 ...
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 ...