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

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Definite integral of a continued fraction function

I came up with this function written as the following continued fraction (please correct me if my notation is incorrect): for $n\in\mathbb{N}$, let $$f(x;n)=x+\operatorname*{\LARGE ...
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1answer
71 views

Evaluating $\int_{0}^{\log\phi}\frac{u^2(e^{2u}+1)}{e^{2u}-1}du$, when $\phi$ is the golden ratio

When I do $$e^u=x+\sqrt{x^2+1}$$ in (17) page 9, see here, after some computations then I obtain $$\frac{\zeta(3)}{10}=\int_{0}^{\log(\phi)}\frac{u^2(e^{2u}+1)}{e^{2u}-1}du,$$ where ...
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1answer
60 views

On $4n+1 = x^2, 5n+1 = y^2$ and the Fibonacci numbers

While answering this question, I decided to look at the particular case, $$4n+1 = x^2\\5n+1 = y^2$$ to be solved simultaneously. The solutions for $n,x,y$ are A157459, A007805, and A049629, ...
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plot golden spiral along with rectangles showing the area split

I can see it's relatively simple to plot the golden spiral using the equation: $$r=e^{b\theta}$$ where $b={{ln(\varphi})\over{\pi\over2}}={{2ln({1+\sqrt5\over2}})\over{\pi}}=0.30634896253$ Here's ...
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113 views

Golden Ratio question

Question: A bar has a rectangular cross sectional area which has dimensions based on the Golden Ratio. A tension force of 500kN was applied on the side of this bar and the other side clamped. This ...
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1answer
30 views

Time Complexity recurrence

When we have the recurrence $T(N)=T(N-1)+T(N-2)$, one normally uses $x^N$ and solves for $x$ which gives the golden ratio. But why does one use $x^N$ and not something else, like $\log(N)$ or $N$?
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3answers
54 views

Is there a pattern to the golden ratio number figures?

The golden ratio or phi is 1.6180339887498948482045... I am wondering if there is a pattern in the numbers so given a certain set of figures, you are able to figure out the rest of the figures ...
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A variation of Viète's formula — an infinite product of nested radicals [duplicate]

Viète's formula expresses an infinite product of nested radicals in terms of $\pi$. Let $$a_1=\sqrt2,\quad a_n=\sqrt{2+a_{n-1}}.$$ Note that $\lim\limits_{n\to\infty}a_n=2.$ Then ...
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1answer
25 views

“Extending” the calculation of the golden ratio using square roots (not silver-ratio)

I'm looking at the following formula: $x =\frac{-n+\sqrt{n^{2}+4n}}{2}$ For $n=1$ this this gives $0.618...$ and then $\frac n x$ gives $1.618...$ which is $\phi$, the golden ratio. What ...
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1answer
58 views

Weakly unimodal function using Golden Section Search

I was going through the Golden Section Search https://en.wikipedia.org/wiki/Golden_section_search and as I understand it should work for every unimodal function. Here, the definition of unimodal ...
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1answer
70 views

Summation of a series involving powers of Fibonacci numbers.

I'm interested in this series: $$\mathcal S_p=\sum_{n=1}^\infty\frac{\left(F_n\right)^p}{2^{np}},\quad p\in\mathbb N,\tag1$$ where $F_n$ are the Fibonacci numbers, defined by the recurrence ...
4
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1answer
64 views

On evaluating the Riemann zeta function, including that $\zeta(2)\gt \varphi$ where $\varphi$ is the golden ratio

A week ago, I got the following : For a positive integer $k$, using Cauchy–Schwarz inequality, $$\left(\sum_{n=1}^{\infty}\frac{1}{n^k(n+1)^k}\right)^2\lt ...
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4answers
2k views

A golden ratio series from a comic book

The eighth installment of the Filipino comic series Kikomachine Komix features a peculiar series for the golden ratio in its cover: That is, $$\phi=\frac{13}{8}+\sum_{n=0}^\infty ...
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In the problem of dividing a line in extreme and mean ratios, how do I show that 1 and x are incommensurable?

In other words, the line is divided at x such that 1/x = x/(1 - x). The problem hints at using the Euclidean algorithm to prove that 1 and x are incommensurable. Also need to show that the proportion ...
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1answer
66 views

Is the golden ratio a transcedental number?

I have been studying the concept of transcedental numbers. Till now, I had taken it for granted that all important numbers like pi and e were transcedental. I have no reason for assuming this or for ...
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1answer
65 views

A different way to calculate number of primes less than a particular number x

According to Wikipedia, series representation of logarithmic integral function is as follows: Now, as per my findings from this we can also calculate number of primes less than a particular number x ...
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1answer
36 views

Proof by induction that *p* = 1/*p*-1 in golden rectangle exercise

The initial rectangle's dimensions is L0 for the length and l0 for the width. A golden rectangle can be obtained when it has the same proportions as the initial rectangle, so p = L0/l0 I am first ...
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0answers
94 views

Is it meaningful in attempting to prove relation between prime numbers and Golden ratio? [closed]

I am trying to define a model which will give number of prime numbers less than or equal to a particular number $N$. I am using golden ratio $\phi$ in this model and I am quite satisfied with my ...
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2answers
67 views

Relationship between golden ratio powers and Fibonacci series

Can anyone prove the following equation? ($F_n$ is the $n$th element of Fibonacci series and $n \in N$.) $\phi = 1 \times \phi + 0$ $\phi^2 = 1 \times \phi + 1 $ $\phi^3 = 2 \times \phi + 1 $ ...
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3answers
101 views

Golden Ratio Approximation

$$\sqrt{1000}-30.0047 \approx \varphi $$ $$[(\sqrt{1000}-30.0047)^2-(\sqrt{1000}-30.0047)]^{5050.3535}\approx \varphi $$ Simplifying Above expression we get ...
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2answers
176 views

a certain simple continued fraction

Given the golden ratio: $$\phi=\frac{1+\sqrt{5}}{2}$$ and the following simple continued fraction: $$G(q,k)=\cfrac{1}{1-q+\cfrac{1}{1-{q^3}^k+\cfrac{1}{1-{q^5}^k+\cfrac{1}{1-{q^7}^k+\ddots}}}}$$ For ...
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1answer
56 views

A proof that $\frac{(2\phi)^n-(-1)^n}{\phi^{2n}-(-1)^n}\cdot\left(2^n-\phi^n\right)\cdot\sqrt5\in\mathbb Q$ for all $n\in\mathbb Z$

During computation of some series (with help of a CAS), at an intermediate step I encountered an expression, that after dropping non-essential parts looks like this:$$\mathcal ...
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228 views

Integral $\int_0^{1/\phi}x\log(x)\log(1+x)\log(1-x)\,dx$

How can we evaluate this definite integral $$I=\int_0^{1/\phi}x\log(x)\log(1+x)\log(1-x)\,dx,$$ where $\displaystyle\phi=\frac{1+\sqrt5}2$ is the golden ratio?
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3answers
166 views

Is $e^{e^{2}}$ a relatively good approximation for $1000\phi$? [closed]

Yesterday night, I found that $e^{e^{2}} \sim 1000\phi$, where $\phi$ is the golden ratio. I believe that it is correct to four decimal places. Would it be considered a relatively good ...
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359 views

Is the Fibonacci lattice the very best way to evenly distribute N points on a sphere? So far it seems that it is the best?

Over in the thread "Evenly distributing n points on a sphere" this topic is touched upon: http://stackoverflow.com/questions/9600801/evenly-distributing-n-points-on-a-sphere. But what I would like to ...
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1answer
81 views

A golden trigonometric diophantine equation

After answering this question I reflected on the identity $$\cos\frac{\pi}{5}=\phi\cos\frac{\pi}{3}$$ and thought of looking for all the quadruplets of positive integers $(a,b,c,d)$ satisfying $$\cos ...
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The $q$-continued fraction for tribonacci constant and others

Let $q = e^{-2\pi}$. We are familiar with Ramanujan's beautiful continued fraction, $$\cfrac{q^{1/5}}{1 + \cfrac{q} {1 + \cfrac{q^2} {1 + \cfrac{q^3} {1+\ddots}}}} = {\sqrt{5+\sqrt{5}\over ...
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proof by induction for golden ratio and fibonacci sequence

I have to prove the following equation by induction for $$x = \phi$$ I am stuck and I don't know how to proceed. This is the equation $$ \phi ^n = f_n\phi + f_{n-1} $$ where $f_n$ is the nth term ...
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3answers
508 views

Simplify $7\arctan^2\varphi+2\arctan^2\varphi^3-\arctan^2\varphi^5$

Let $\varphi=\frac{1+\sqrt5}2$ (the golden ratio). How can I simplify the following expression? $$7\arctan^2\varphi+2\arctan^2\varphi^3-\arctan^2\varphi^5$$
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1answer
124 views

How to find the center of a log spiral?

Given just a few points on a log spiral, how to find the center? Considering that the circle is a degenerate case of the log spiral, is there a way to generalize the method for finding circle centers ...
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1answer
94 views

Mathematical importance of the golden ratio [duplicate]

I know the golden ratio is the limit of the ratios of consecutive Fibonacci numbers and that it appears when studying many related combinatorial objects (such as the sequences of zeros and ones with ...
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4answers
321 views

How to compute $\int_0^\infty \frac{1}{(1+x^{\varphi})^{\varphi}}\,dx$?

How to compute the integral, $$\int_0^\infty \frac{1}{(1+x^{\varphi})^{\varphi}}\,dx$$ where, $\varphi = \dfrac{\sqrt{5}+1}{2}$ is the Golden Ratio?
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1answer
112 views

Even Fibonacci Numbers and $\sqrt{5}$

My question is simple, but a mystery to me. Skip to the last paragraph if you're not interested in the story of my exploration. EDIT: I seem to have misinterpreted a key detail regarding how the ...
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4answers
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Proof the golden ratio with the limit of Fibonacci sequence [duplicate]

Let $F_n=F_{n-1}+F_{n-2}$ the Fibonacci numbers, and $\phi=\frac{1+\sqrt5}{2}$ The exercise asks me to prove that: $\lim\limits_{n \to \infty}\frac{F_{n+1}}{F_n}=\phi$. Sorry as can be proceed??
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Compare $\sum_{k=1}^n \left\lfloor \frac{k}{\varphi}\right\rfloor$ …

Given two integer sequences \begin{equation*} \displaystyle A_n=\sum_{k=1}^n \left\lfloor \frac{k}{\varphi}\right\rfloor , \end{equation*} \begin{equation*} ...
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1answer
36 views

$\phi$, and the uses of an alternate formula

I was trying to find the solution to the formula: $$x = \sum_{n=1}^\infty{x^{-n}}$$ I found it to be the golden ratio, or $\phi = \frac{1 + \sqrt{5}}{2}$. I do not know if this has already been ...
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Can't find ANY golden ratio in the schroder house…

The Schroder House (The Netherlands) is supposed to be designed using the "golden ratio". I'm having trouble finding these golden ratio's. A lot of rectangles, windows, house sections, etc. appear to ...
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2answers
65 views

Fibonacci recursive algorithm yields interesting result

After writing a program in Java to generate Fibonacci numbers using a recursive algorithm, I noticed the time increase in each iteration is approximately $\Phi$ times greater than the previous. ...
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1answer
83 views

$ab-(a+b)(a-b)=0$ and the Golden ratio.

I have found: $b=a*\phi$ $b=a*(-\phi)$ $b=a/\phi$ Trying to find the correlation with the equation and phi, any insight how to demonstrate this or a proof?
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Coefficients of the polynomials generated by $f_0=x,\ f_{i+1}=f_i\pm\dfrac1{f_i}$.

Let $f_0=x,\ f_{i+1}=f_i\pm\dfrac1{f_i}$ for $i\geq0$, i.e., $f_i=\dfrac{\sqrt{f_{i+1}^2\mp4}+f_{i+1}}2$ I have observed that $f_1=\dfrac{x^2\pm1}x$ $f_2=\dfrac{x^4\pm3x^2+1}{x(x^2\pm1)}$ ...
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2answers
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Why is it that $\frac{\sin 30}{\sin 18}$ is equal to the golden ratio?

If you calculate $\frac{\sin 30}{\sin 18}$, where $18$ and $30$ are in degrees, the result is $\phi$, or alternately $\frac{1 + \sqrt{5}}{2}$. I know that these numbers add up, but is there any ...
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734 views

Why does every “fibonacci like” series converge to $\phi$?

It's is well known that the ratio of side-by-side fibonacci numbers converge to $\phi$. But it seems by my calculations, that if one starts with any pair of numbers one will also get a ratio that ...
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2answers
133 views

Why is the radian golden angle $(1-1/\varphi)\cdot2\pi=\pi(3-\sqrt5)\approx2.39996$ so close to a 'nice' rational number?

I was reading about phytollaxis in plants and Fermat's spirals when I came across the Wikipedia article on golden angles. Surprisingly, the radian golden angle is very nearly approximated by a simple ...
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3answers
256 views

Inequality of the Fibonacci sequence and the golden ratio

How can I prove that for each $n\in\Bbb Z^+$ $$\frac{f_{2n}}{f_{2n-1}}\leq\frac{1+\sqrt{5}}{2}$$ where each $f_i$ is a term of the Fibonacci sequence. Any help is really appreciated
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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
555 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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83 views

Why does the tribonacci constant have a trilogarithm ladder?

When I came across the dilogarithm ladders of Coxeter and Landen, namely, $$\text{Li}_2(\alpha^6)-4\text{Li}_2(\alpha^3)-3\text{Li}_2(\alpha^2)+6\text{Li}_2(\alpha)-\tfrac{7}{5}\zeta(2)=0\tag1$$ ...
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2answers
145 views

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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2answers
2k 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 ...