Question

Is there something special about $x^2-x-1$?

score 1.6k · Answer 1 · 2017-03-24 20:26:08

1.6k

Views

Is there something special about $x^2-x-1$?

Published on 24 Mar 2017 - 20:26

score 150 · Answer 2 · 2017-03-25 00:14:46

150

Views

How to prove that $a_{n+1} = 1 + \frac{a_n}{a_n +1}$ is monotone increasing with $a_1=1$

Published on 25 Mar 2017 - 0:14

score 646 · Answer 3 · 2017-03-28 08:24:04

646

Views

Find $q$ such that $[q[qn]]+1=[q^2n]$ for $n=1,2,\dotsc$

Published on 28 Mar 2017 - 8:24

score 103 · Answer 4 · 2017-04-01 20:40:41

103

Views

How can we demonstrate that $\sqrt{\phi}=5^{1/4}\cos\left({1\over 2}\tan^{-1}(2)\right)?$

Published on 01 Apr 2017 - 20:40

score 148 · Answer 5 · 2023-02-08 10:40:27

148

Views

Pole of the golden spiral

Published on 08 Feb 2023 - 10:40

score 440 · Answer 6 · 2023-02-09 17:23:05

440

Views

Relation between $\phi$ and $\pi$

Published on 09 Feb 2023 - 17:23

score 83 · Answer 7 · 2023-02-12 20:21:00

83

Views

Why is $\phi^x=\underset{n\to\infty}{\lim} \frac{F_n}{F_{n-x}}$?

Published on 12 Feb 2023 - 20:21

score 133 · Answer 8 · 2023-02-21 16:24:00

133

Views

A connection between Fibonacci numbers and the golden ratio

Published on 21 Feb 2023 - 16:24

score 80 · Answer 9 · 2026-03-25 16:03:33

80

Views

What is the exact value of $\sum\limits_{n=1}^\infty \frac{1}{F_n}$?

Published on 25 Mar 2026 - 16:03

score 53 · Answer 10 · 2023-03-08 12:42:59

53

Views

For Which $a$ In $[0,1]^n$ Is $\text{frac}(ta)$ Maximally Aperiodic

Published on 08 Mar 2023 - 12:42

score 884 · Answer 11 · 2023-03-24 05:52:02

884

Views

Is there any method other than Feynman’s trick which can deal further with powers higher than 2?

Published on 24 Mar 2023 - 5:52

score 94 · Answer 12 · 2023-03-25 13:07:33

94

Views

Show an integral inequality $\int_{0}^{\phi}f\left(x^{\phi}\right)+f\left(\phi^{\phi}-x\right)dx<\phi$

Published on 25 Mar 2023 - 13:07

score 251 · Answer 13 · 2023-04-10 18:09:00

251

Views

Why does there seem to be a relation between the fifth roots of $32$ and $\varphi$ (the golden ratio)?

Published on 10 Apr 2023 - 18:09

score 332 · Answer 14 · 2023-04-25 14:52:45

332

Views

If $a^2-a-1=0$ where $a\gt0$, then what does $a^6$ equal? (Olympiad question)

Published on 25 Apr 2023 - 14:52

score 79 · Answer 15 · 2026-03-25 07:45:43

79

Views

Different way of proving: $\sum_{n=1}^{\infty}\frac{\left(16(-1)^n+5\right)\left(\phi H_n+\frac{1}{n^3}\right)+\frac{11}{n^3}}{n^2}=\zeta(5)$

Published on 25 Mar 2026 - 7:45

List Question

Is there something special about $x^2-x-1$?

How to prove that $a_{n+1} = 1 + \frac{a_n}{a_n +1}$ is monotone increasing with $a_1=1$

Find $q$ such that $[q[qn]]+1=[q^2n]$ for $n=1,2,\dotsc$

How can we demonstrate that $\sqrt{\phi}=5^{1/4}\cos\left({1\over 2}\tan^{-1}(2)\right)?$

Pole of the golden spiral

Relation between $\phi$ and $\pi$

Why is $\phi^x=\underset{n\to\infty}{\lim} \frac{F_n}{F_{n-x}}$?

A connection between Fibonacci numbers and the golden ratio

What is the exact value of $\sum\limits_{n=1}^\infty \frac{1}{F_n}$?

For Which $a$ In $[0,1]^n$ Is $\text{frac}(ta)$ Maximally Aperiodic

Is there any method other than Feynman’s trick which can deal further with powers higher than 2?

Show an integral inequality $\int_{0}^{\phi}f\left(x^{\phi}\right)+f\left(\phi^{\phi}-x\right)dx<\phi$

Why does there seem to be a relation between the fifth roots of $32$ and $\varphi$ (the golden ratio)?

If $a^2-a-1=0$ where $a\gt0$, then what does $a^6$ equal? (Olympiad question)

Different way of proving: $\sum_{n=1}^{\infty}\frac{\left(16(-1)^n+5\right)\left(\phi H_n+\frac{1}{n^3}\right)+\frac{11}{n^3}}{n^2}=\zeta(5)$

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