Questions tagged [golden-ratio]

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

Filter by
Sorted by
Tagged with
0 votes
2 answers
51 views

Proof that $\sum_{n=1}^\infty\frac{F_n}{3^n n} = \frac{\ln(\phi+1)}{\sqrt{5}}$

I conjectured by computation the following, but I’m not sure where to start to prove it. $$\sum_{n=1}^\infty\frac{F_n}{3^n n} = \frac{\ln(\phi+1)}{\sqrt{5}}$$ where $F_n$ are the Fibonacci numbers. I’...
user avatar
  • 3,195
0 votes
0 answers
34 views

What's the intuition of the relation between fibonacci-like sequences and the proportion used to obtain the golden ratio?

Everyone knows that we can obtain the golden ratio from the following proportion: $$\frac{a}{b} = \frac{a+b}{a}$$ We also know that we get ${\phi}^N$ when we try to find a function that satisfies the ...
user avatar
  • 1
2 votes
1 answer
46 views

Rational approximations of the golden ratio: how to prove this limit exists?

Given a positive real number $\alpha$ and a positive rational number $p/q$ in reduced form let's define the quality of $p/q$ as an approximation to $\alpha$ as$$-\log_q|\alpha - p/q|$$ I'm looking at ...
user avatar
1 vote
0 answers
74 views

proof fibonacci sequence is small o(2^n) without using closed formula

I need to prove that for the given fibonacci sequence, with initial values: f(1)=1 f(2)=2 f(n)=f(n-1)+f(n-2) f(n) is belong to small o(2^n). I need to prove it without using the closed formula of ...
user avatar
6 votes
1 answer
76 views

Why does the golden ratio emerge in this primorial-related sequence?

Let $$f(i):=\left\lfloor\frac{p_i\#}{\varphi(p_i\#)}\right\rfloor,$$ where $p_i$ is the $i$th prime, $\#$ is the primorial operator, and $\varphi$ is totient. Example $$f(3)=\left\lfloor\frac{5\#}{\...
user avatar
  • 4,941
0 votes
2 answers
45 views

Compute $p_{n+1} = p^2 \frac{r_1^n-r_2^n}{r_1-r_2}, n \geq 0$

I need to compute $$p_{n+1} = p^2 \frac{r_1^n-r_2^n}{r_1-r_2}, n \geq 0$$ where $p=\frac{1}{2}$, $r_1= \frac{1+\sqrt5}{4}= \frac{1}{2}\varphi$ and $r_2= \frac{1-\sqrt5}{4}$. Using properties of ...
user avatar
0 votes
0 answers
31 views

Non-periodic continued fraction with explicitly known convergents?

Is an irrational number with non-periodic continued fraction expansion known, for which one can give explicit formulas for the convergents $p_n/q_n$ or at least for the denominators $q_n$ (similar to ...
user avatar
1 vote
1 answer
54 views

Relation between logarithmic spirals and the golden ratio

The polar equation of a logarithmic spiral curve is given by $$ r= ae^{b\theta} $$ where each point on the curve is described in polar coordinates: $r$ is the distance from the origin and $\theta$ is ...
user avatar
  • 1,354
1 vote
0 answers
77 views

What does the golden ratio have to do with complex hyperbola and real circle

If you have $xy = i $ and $x^2 + y^2 = 1$ then you get the solutions that have the golden ratio in them. These are the solutions Wolfram calculation: https://www.wolframalpha.com/input/?i=xy%3Di%2C+x%...
user avatar
7 votes
6 answers
365 views

Is my proof for the Irrationality of the Golden Ratio correct?

I am trying to make a proof about the irrationality of the golden ratio by contradiction. I substituted the fraction $\frac{p}{q}$ for $\varphi$, where p and q are integers that don't have a common ...
user avatar
0 votes
0 answers
45 views

L and M are midpoints of equilateral triangle ABC, and LM meets the circumcircle of the triangle at Y. Prove LM/MY is the golden ratio.

So far I've tried to split the triangle into smaller 30-60-90 triangles, and letting LM = x make some proportions and see if they can be related in anyway. I'm not sure what to do about MY though. Any ...
user avatar
3 votes
1 answer
70 views

Dodecahedron and golden ratio algebra

We can see that the volume of a dodecahedron of size $2\varphi$, where $\varphi=\frac{\sqrt{5}-1}{2}$, can be found in two ways. The first one uses the pentagonal pyramids with the faces as basis and ...
user avatar
0 votes
1 answer
47 views

Equations for half-integer points on generalized complex Fibonacci sequence (metallic mean sequence)

I have been experimenting with generalizing the Fibonacci sequence, and Fibonacci-like "metallic mean" sequences such as the Pell sequence, to non-integer and complex values. The standard, ...
user avatar
  • 751
2 votes
2 answers
153 views

Golden ratio in complex number squares

In the Argand diagram shown below the complex numbers $– 1 + i, 1 + i, 1 – i, – 1 – i$ represent the vertices of a square ABCD. The equation of its diagonal BD is $y = x$. The complex number $k + ki$ ...
user avatar
  • 1,073
1 vote
0 answers
124 views

Exercise 0.21 in Miles Reid's Commutative Algebra

I have the following exercise in Miles Reid's Undergraduate Commutative Algebra: Exercise 0.21: Consider the ring $B′=\Bbb Z[\tau]$, where $\tau^2=\tau+1$. Show that an element $a+b\tau$ is a unit of ...
user avatar
0 votes
1 answer
35 views

For which recursive functions (or infinite expressions) can substitution be used for a solution?

I have a question about recursive functions and their convergence as sequences. We know that the golden ratio can be represented by: $$ \phi= 1+\frac{1}{1+\frac{1}{1+\frac{1}{...}}} $$ Rewriting the ...
user avatar
0 votes
2 answers
94 views

why is the golden ratio algebraic and the golden angle transcendental [closed]

I'm referencing the golden angle from here https://en.wikipedia.org/wiki/Golden_angle, where it's defined as $ 2\pi(1-\frac{1}{\varphi})$. All constructible numbers are algebraic numbers. A ...
user avatar
8 votes
2 answers
184 views

How to solve $\left(\sqrt{\frac{x-1}{x}}\right)^{x^2}=\left(\frac{1}{x}\right)^{x+1}$?

I need help to solve this equation, please. $$\left(\sqrt{\frac{x-1}{x}}\right)^{x^2}=\left(\frac{1}{x}\right)^{x+1}$$ I know that the solution is $x=\varphi$ (the golden ratio). I got this result by ...
user avatar
  • 109
3 votes
2 answers
111 views

Is there a function for $[1;x,x,...]$ (as an infinite continued fraction)?

So I was experimenting with continued fractions and came up with an idea for a function $$f(x) =1+\cfrac{1}{x+\cfrac{1}{x+\cfrac{1}{x+\cfrac{1}{x+\cfrac{1}{x+\cfrac{1}{x+\dots}}}}}},$$ as an infinite ...
user avatar
  • 403
0 votes
0 answers
28 views

How are Ostrowski Numeration Systems built from a periodic continued fraction?

Ostrowski number systems represent integers and real numbers using the denominators of the convergents of continued fractions. One better known special case of this is Zeckendorf's Theorem and related ...
user avatar
  • 1,036
10 votes
1 answer
493 views

A Golden Angle Conjecture

Update: this conjecture is broken, for n=7. See final note. This conjecture is related to the process of phyllotaxis in plants, and understanding why nature would choose to iterate the Golden Angle ...
user avatar
0 votes
0 answers
62 views

Julia Set Fixed Points And The Golden Ratio

When calculating fixed points for the Basilica Julia set ($z_{n+1}=z^2+c$, $c=-1$), the fixed points are given by $$ z^*_0(-1)=\frac{1 \mp \sqrt{1-4(-1)}}{2}=\frac{1 \mp \sqrt{5}}{2} $$ This is ...
user avatar
1 vote
3 answers
48 views

Fibonacci golden ratio question without assuming convergence a priori

Prove $\frac{F(n+1)}{F(n)}$ converges to $\phi$ without assuming a priori that it converges If I know it converges, then I know it converges to $\phi$ Since, if $\lim_{n\to\infty} \frac{F(n+1)}{F(n)} \...
user avatar
  • 380
1 vote
0 answers
108 views

Approximation of the golden ratio using exponential

Claim : $$\left(1+e^{-1}\right)^{3}\cdot\left(1-e^{-1}\right)-\frac{\left(\sqrt{5}+1\right)}{2}<0$$ I found it accidentaly using Desmos .It's quite good as approximation of the golden ratio. Some ...
user avatar
1 vote
1 answer
49 views

Rectangles with side lengths proportional to transcendental numbers BESIDES $\varphi$

A rectangle with side lengths proportional to $\varphi$ is a "golden rectangle". If we perform a procedure of subtracting the shorter side length from the longer side length, a golden ...
user avatar
0 votes
0 answers
68 views

Derive an exact value for the Golden ratio

Have I accurately solved with the given information?
user avatar
  • 1,242
2 votes
2 answers
77 views

Why does the golden ratio show up as the solution to these two simultaneous equations?

I was given the following simultaneous equations to solve on a homework sheet: $$ x^2 + y^2 = 3\\x-y=1 $$ And when I did so I got the answers of: $$ (\varphi,-1/\varphi)\\ (1/\varphi,-\varphi) $$ I ...
user avatar
  • 57
8 votes
2 answers
102 views

Is there a notion of Lucas/Fibonacci sequences beyond adding two numbers at a time, if so where could I read more about that?

So I recently learned about lucas/fibbonaci sequences EG; 1,1,2,3,5,7 or 3,4,7,11,18... A fact I learned about these sequence is that the limit of the quotient of the terms approaches the golden ratio ...
user avatar
2 votes
2 answers
82 views

Estimating golden ration by simulation

I've been looking at Monte Carlo simulation recently, and have been using it to approximate constants such as π (circle inside a rectangle, proportionate area). However, I'm unable to think of a ...
user avatar
0 votes
2 answers
93 views

Fibonacci generating function - quick way to see formula

I know that $$ \frac{1}{1-z-z^2} = \sum_{k = 1}^\infty a_nz^n $$ in $B_{\varepsilon}(0) \subseteq \mathbb{C}$ where $(a_n)_{n \in \mathbb{N}}$ denotes the Fibonacci series and $\varepsilon > 0$ is ...
user avatar
  • 4,614
1 vote
3 answers
239 views

Improper integral inequality including the golden ratio and the Sophomore's dream

It's an inequality I found nice let me propose it : Let $0\leq x$ then we have : $$\int_{0}^{\infty}\sin\left(x^{-x}\right)dx<\phi=\frac{\left(1+\sqrt{5}\right)}{2}\quad (I)$$ My attempt : First ...
user avatar
0 votes
0 answers
38 views

Bounding the sum of a floor function involving the golden ratio

Consider the sum \begin{equation*} S_n = \displaystyle \sum_{k=1}^n \left( k\varphi - \left\lfloor k\varphi\right\rfloor -\frac12 \right) \end{equation*} where $\quad\varphi=\dfrac{1+\sqrt{5}}{2}\quad$...
user avatar
0 votes
0 answers
66 views

Exercise 0.20 in Miles Reid's Commutative Algebra

I have the following exercise in Miles Reid's Undergraduate Commutative Algebra: Exercise 0.20: Consider the ring $B'=\Bbb Z[\tau]$, where $\tau=\frac{1+\sqrt 5}{2}$ is the so called "golden ...
user avatar
1 vote
2 answers
90 views

$a^2\phi=2$ , what is the value of $2a(\phi+1)$?

I reduced a problem to this: We have $a^2\phi=2$ where $a>0$ what is the value of $2a(\phi+1)$ ? $1)2\sqrt{2\sqrt5+4}\qquad\qquad2)2\sqrt{\sqrt5+4}\qquad\qquad3)2\sqrt{2\sqrt5+1}\qquad\qquad4)2\...
user avatar
  • 5,543
14 votes
1 answer
271 views

Prove that $\int _{-\infty }^{+\infty }{\frac {\mathrm {d} z}{(\phi ^{n}z)^{2}+(F_{2n+1}-\phi F_{2n})(e^{\gamma }z^{2}+\zeta (3)z-\pi )^2}}=1$

Recently while dealing with few interesting integrals, I was quite fascinated by this one: $$\int _{-\infty }^{+\infty }{\dfrac {1}{(\phi ^{n}z)^{2}+(F_{2n+1}-\phi F_{2n})(e^{\gamma }z^{2}+\zeta (3)z-\...
user avatar
1 vote
1 answer
77 views

Prove without induction that the sequence $a_{n}=1+\frac{1}{a_{n-1}}$ $=$ $\frac{F_{n+3}}{F_{n+2}}$ with $a_1=2$ where $F_n$ is the Fibonacci sequence

While trying to prove that the sequence $a_{n+1}=1+\frac{1}{a_n}$ with $a_1=2$ converges to Phi(the second sequence obviously converges to Phi), I recognized the pattern when I calculated the first ...
user avatar
  • 100
0 votes
2 answers
69 views

Proof if the digits of the golden ratio are in sequence in the digits of Pi or not [closed]

Since the digits of $\pi$ are uniformly random and infinite, any finite sequence of digits can be found in sequence somewhere among the digits of $\pi$. But does this also holds when it comes to ...
user avatar
  • 113
0 votes
0 answers
38 views

How deep into a Fibonacci matrix would be needed to calculate pi to 100 digits?

(0,1),1,2,3 (1,1),2,3,5 (2,1),3,4,7 (3,1),4,5,9 Then use Wallis product, as a wave descending the layers, to calculate pi in various ways such as pi/2 = 2/1 x 2/3 x 4/3 x 4/5 x... Also able to ...
user avatar
1 vote
0 answers
55 views

Is it true that $f(x)^{f(x)^{\dots}}$ with $f(x) = -x^2 -x+1$ converges to the Golden Ratio? Why?

My son accidentally discovered and, given $f(x)=-x^{2}-1x+1$, $f\left(x\right)^{f\left(x\right)^{f\left(x\right)^{f\left(x\right)}}}$ intersects 1.0 (i.e., $y=1.0$) at the negative of the golden ratio,...
user avatar
3 votes
1 answer
114 views

iterated function $a(n) = \lfloor n\phi + 0.5\rfloor$

Let $a(n) = \lfloor n\phi + 0.5\rfloor$ for all $n \geqslant 1$, where $\phi$ is the golden ratio. Now let $$a(n)^k = a(a(\ldots(a(n))))$$ where we have iterated $a(n)$, $k$ times in the RHS. I am ...
user avatar
2 votes
0 answers
46 views

Fibonacci number of local minima in sum of sines

I have this function $$\begin{align}f_N:[-1,1]&\longrightarrow [-1,1] \\ \hspace{-10mm}x &\longrightarrow \frac{1}{N}\sum_{i=0}^N cos(\phi^i \pi x) \end{align}$$ where $\phi=\frac{1+\sqrt 5}{2}...
user avatar
0 votes
2 answers
104 views

Golden Exponent? Tetration

I read somebody say “golden exponent $ x^x=x+1 $” now I didn’t understand what he meant but it really fascinated me thinking about some type of tetration version of the golden number. A number with a ...
user avatar
4 votes
0 answers
86 views

Given a recursion $a_{n+ 1}= a_{n}^{2}+ 1$ with $a_{0}> 0,$ I want to consider the asymptotic behavior of this sequence.

Given a recursion $a_{n+ 1}= a_{n}^{2}+ 1$ with $a_{0}> 0,$ I want to consider the asymptotic behavior of this sequence. Till now $$a_{n}> a_{n- 1}^{2}> \cdots> a_{0}^{2^{n}}$$ and for $\...
user avatar
2 votes
4 answers
214 views

Prove that : $ \int_{0}^{2\ln{\varphi}}{\theta\ln{\left(2\sinh{\frac{\theta}{2}}\right)}\,\mathrm{d}\theta}=-\frac{1}{5}\zeta\left(3\right) $

Denoting $ \varphi=\frac{1+\sqrt{5}}{2}=\mathrm{Golden\ Ratio} $. How would you prove that : $$ \int_{0}^{2\ln{\varphi}}{\theta\ln{\left(2\sinh{\frac{\theta}{2}}\right)}\,\mathrm{d}\theta}=-\frac{1}{...
user avatar
  • 7,680
1 vote
0 answers
62 views

Approximation to the sum of reciprocals of prime numbers squared involving $\phi$, the golden ratio

I was interested in the prime zeta function and its values, so I calculated in Excel the sum of reciprocals of prime numbers squared up to $1$ MM, and tried to relate it to some known irrational ...
user avatar
1 vote
1 answer
67 views

Why is the intersections between $f(x)=x^{2}+x^{-2}-3$ and the x-axis resembles $\varphi$?

The story behind this is quite silly, I was messing around with functions in geogebra, inputting several functions to see their properties, until I was amazed by something i don't know about. ...
user avatar
  • 131
3 votes
3 answers
81 views

Prove that $\left \lfloor \frac{1+\lfloor na+1/a\rfloor}{a} \right \rfloor=n$

If $a \geq \frac{1+\sqrt{5}}{2}$ and $n \in \Bbb W$, prove that $$\left \lfloor \frac{1+\left\lfloor \frac{1+na^2}{a}\right\rfloor}{a} \right \rfloor=n.$$ I could prove only when $a$ is an integer, ...
user avatar
0 votes
1 answer
222 views

golden section search algorithm: why are there two 'update' equations?

I understand that the golden section search algorithm (for finding minimum points) is loosely based on the bisection method (for finding roots). In both methods, we first assign an upper and a lower ...
user avatar
  • 1,608
1 vote
2 answers
326 views

$\pi$ & $\phi$ (Golden ratio), Pentagon inscribed in unit circle

Everyone is aware that square inscribed in unit circle and infinite product giving rise to $\pi$. One of the simplest way to represent $\pi$ with the help of nested radical as follows $$\pi = \lim_{n\...
user avatar
2 votes
0 answers
79 views

Nested radicals, Golden ratio, number series

Finite and infinite expansion of nested Radical involving Golden ratio as follows $2\cos\frac{2\pi}{5} = \frac{1}{\phi} $ where $\phi$ is Golden ratio $(2\cos72°)$ Let us expand this with increasing ...
user avatar

1
2 3 4 5
10