Question

$\int_{0}^{1}{(1-x)(1-2x^{\phi})+\phi(x-x^{\phi})\over (1-x)^2}\cdot{\left(1-x^{\phi}\over 1-x\right)^{1\over \phi}}\mathrm dx=\phi^{\phi}$

score 494 · Answer 1 · 2017-01-24 10:02:23

494

Views

$\int_{0}^{1}{(1-x)(1-2x^{\phi})+\phi(x-x^{\phi})\over (1-x)^2}\cdot{\left(1-x^{\phi}\over 1-x\right)^{1\over \phi}}\mathrm dx=\phi^{\phi}$

Published on 24 Jan 2017 - 10:02

score 107 · Answer 2 · 2017-01-30 14:07:46

107

Views

Confusion regarding Golden Ratio.

Published on 30 Jan 2017 - 14:07

score 86 · Answer 3 · 2026-03-25 21:20:52

86

Views

Constants connection of $\gamma$ and $\phi$

Published on 25 Mar 2026 - 21:20

score 107 · Answer 4 · 2017-02-03 06:09:15

107

Views

Need to show $x_1=x_2=x_3=\ldots=x_n=\phi$ for the given sequence of $n$ positive positive numbers.

Published on 03 Feb 2017 - 6:09

score 384 · Answer 5 · 2017-02-05 07:43:47

384

Views

How can we show that $4\arctan\left({1\over \sqrt{\phi^3}}\right)-\arctan\left({1\over \sqrt{\phi^6-1}}\right)={\pi\over 2}$

Published on 05 Feb 2017 - 7:43

score 209 · Answer 6 · 2026-03-25 19:10:28

209

Views

How can we show that :- $\sum_{n=1}^{\infty}\left({\phi^2+\gamma\over n+1}-{\phi+\gamma\over n}-\ln{n+1\over n}\right)=-\phi^2$

Published on 25 Mar 2026 - 19:10

score 248 · Answer 7 · 2017-02-20 11:52:15

248

Views

Machin's formula terms expressed in term of golden ratio : $4\cot^{-1}{5}-\cot^{-1}{239}={\pi\over 4}$

Published on 20 Feb 2017 - 11:52

score 159 · Answer 8 · 2017-02-22 00:45:43

159

Views

Math Competition Question

Published on 22 Feb 2017 - 0:45

score 444 · Answer 9 · 2017-02-23 07:58:30

444

Views

How does one show that $\int_{0}^{1}{1\over 1+\phi x^4}\cdot{\mathrm dx\over \sqrt{1-x^2}}={\pi\over 2\sqrt{2}}?$

Published on 23 Feb 2017 - 7:58

score 232 · Answer 10 · 2017-02-24 09:29:26

232

Views

Q: How does one show that $\int_{0}^{1}{1-x^{\sqrt5}\over (1-x^{\phi})^{\phi}}\mathrm dx={\phi^2}?$

Published on 24 Feb 2017 - 9:29

score 278 · Answer 11 · 2017-02-25 09:04:54

278

Views

Prove that $\phi=\frac{13}{8} + \sum \limits_{n=0}^{\infty}\frac{(-1)^{(n+1)}(2n+1)!}{n!\,(n+2)!\,4^{(2n+3)}}$.

Published on 25 Feb 2017 - 9:04

score 2.1k · Answer 12 · 2017-02-25 19:54:12

2.1k

Views

How many pairs of numbers are there so they are the inverse of each other and they have the same decimal part?

Published on 25 Feb 2017 - 19:54

score 488 · Answer 13 · 2017-02-26 06:38:23

488

Views

Why do these two graphs intersect at 0.618 (golden ratio)?

Published on 26 Feb 2017 - 6:38

score 99 · Answer 14 · 2017-03-15 14:54:47

99

Views

Looking for a closed form of a polylog expression involving the golden ratio

Published on 15 Mar 2017 - 14:54

score 92 · Answer 15 · 2017-03-23 17:06:46

92

Views

Golden Ratio example not satisfying ratio

Published on 23 Mar 2017 - 17:06

List Question

$\int_{0}^{1}{(1-x)(1-2x^{\phi})+\phi(x-x^{\phi})\over (1-x)^2}\cdot{\left(1-x^{\phi}\over 1-x\right)^{1\over \phi}}\mathrm dx=\phi^{\phi}$

Confusion regarding Golden Ratio.

Constants connection of $\gamma$ and $\phi$

Need to show $x_1=x_2=x_3=\ldots=x_n=\phi$ for the given sequence of $n$ positive positive numbers.

How can we show that $4\arctan\left({1\over \sqrt{\phi^3}}\right)-\arctan\left({1\over \sqrt{\phi^6-1}}\right)={\pi\over 2}$

How can we show that :- $\sum_{n=1}^{\infty}\left({\phi^2+\gamma\over n+1}-{\phi+\gamma\over n}-\ln{n+1\over n}\right)=-\phi^2$

Machin's formula terms expressed in term of golden ratio : $4\cot^{-1}{5}-\cot^{-1}{239}={\pi\over 4}$

Math Competition Question

How does one show that $\int_{0}^{1}{1\over 1+\phi x^4}\cdot{\mathrm dx\over \sqrt{1-x^2}}={\pi\over 2\sqrt{2}}?$

Q: How does one show that $\int_{0}^{1}{1-x^{\sqrt5}\over (1-x^{\phi})^{\phi}}\mathrm dx={\phi^2}?$

Prove that $\phi=\frac{13}{8} + \sum \limits_{n=0}^{\infty}\frac{(-1)^{(n+1)}(2n+1)!}{n!\,(n+2)!\,4^{(2n+3)}}$.

How many pairs of numbers are there so they are the inverse of each other and they have the same decimal part?

Why do these two graphs intersect at 0.618 (golden ratio)?

Looking for a closed form of a polylog expression involving the golden ratio

Golden Ratio example not satisfying ratio

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