User achille hui

97 votes

Accepted

Conjecture $\int_0^1\frac{\mathrm dx}{\sqrt{1-x}\ \sqrt[4]x\ \sqrt[4]{2-x\,\sqrt3}}\stackrel?=\frac{2\,\sqrt2}{3\,\sqrt[8]3}\pi$

answered Oct 31, 2013 at 7:42

72 votes

Accepted

Is $0.1010010001000010000010000001 \ldots$ transcendental?

transcendental-numbers decimal-expansion

answered May 5, 2014 at 4:27

68 votes

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Integral $\int_0^1\frac{\ln\left(x+\sqrt2\right)}{\sqrt{2-x}\,\sqrt{1-x}\,\sqrt{\vphantom{1}x}}\mathrm dx$

calculus integration definite-integrals logarithms closed-form

answered Mar 15, 2014 at 9:46

65 votes

What are some examples of mathematics that had unintended useful applications much later?

math-history big-list applications

answered Sep 7, 2013 at 22:59

62 votes

Does non-symmetric positive definite matrix have positive eigenvalues?

linear-algebra matrices eigenvalues-eigenvectors positive-definite

answered Mar 9, 2013 at 10:22

55 votes

Accepted

Conjecture $\int_0^1\frac{dx}{\sqrt[3]x\,\sqrt[6]{1-x}\,\sqrt{1-x\left(\sqrt{6}\sqrt{12+7\sqrt3}-3\sqrt3-6\right)^2}}=\frac\pi9(3+\sqrt2\sqrt[4]{27})$

calculus integration closed-form conjectures hypergeometric-function

answered Feb 28, 2014 at 12:19

53 votes

Find $xy+yz+zx$ given systems of three homogenous quadratic equations for $x, y, z$

algebra-precalculus polynomials contest-math systems-of-equations

answered Feb 9, 2018 at 12:20

52 votes

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Why does Wolframalpha think that this sum converges?

real-analysis convergence-divergence summation

answered Mar 1, 2016 at 12:37

49 votes

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How does one [easily] calculate $\sum\limits_{n=1}^\infty\frac{\mathrm{pop}(n)}{n(n+1)}$?

summation

answered Jun 29, 2013 at 13:30

47 votes

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Prove that there exists a triangle which can be cut into 2005 congruent triangles.

geometry contest-math

answered Nov 26, 2018 at 2:57

47 votes

Accepted

Is there a fast way to prove a tridiagonal matrix is positive definite?

matrices positive-definite tridiagonal-matrices

answered May 11, 2018 at 16:46

46 votes

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Evaluating $\int_{0}^{1}\cdots\int_{0}^{1}\bigl\{\frac{1}{x_{1}\cdots x_{n}}\bigr\}^{2}\:\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}$

calculus integration multivariable-calculus definite-integrals euler-mascheroni-constant

answered Aug 17, 2014 at 4:16

39 votes

A strange integral: $\int_{-\infty}^{+\infty} {dx \over 1 + \left(x + \tan x\right)^2} = \pi.$

calculus integration definite-integrals improper-integrals

answered Nov 11, 2014 at 13:23

39 votes

Conflicting limit answers using calculator and wolfram alpha

calculus wolfram-alpha calculator

answered Jun 11, 2015 at 4:50

39 votes

Can you complete the expression $2 \underline{ }\, \underline{ }\, \underline{ } \,\underline{ } 5 = 2015$?

elementary-number-theory recreational-mathematics puzzle

answered Aug 4, 2015 at 22:19

36 votes

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Is Tolkien's Middle Earth flat?

geometry ring-theory noneuclidean-geometry

answered Jul 8, 2015 at 12:26

36 votes

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Find large power of a non-diagonalisable matrix

linear-algebra matrices exponentiation diagonalization

answered Aug 21, 2016 at 7:27

35 votes

Geometry problem involving infinite number of circles

sequences-and-series geometry circles

answered Aug 20, 2014 at 16:50

35 votes

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How prove this matrix $\det (A)=\left(\frac{1}{\ln{(a_{i}+a_{j})}}\right)_{n\times n}\neq 0$

linear-algebra matrices determinant

answered Nov 23, 2013 at 10:35

34 votes

Accepted

A very different property of primitive Pythagorean triplets: Can number be in more than two of them?

algebra-precalculus number-theory elementary-number-theory diophantine-equations pythagorean-triples

answered Dec 9, 2016 at 11:37

32 votes

What's the limit of $\sqrt{2 + \sqrt{2-\sqrt{2+\sqrt{2-\sqrt{2-\sqrt{2 + ...}}}}}} $?

calculus real-analysis limits closed-form nested-radicals

answered May 8, 2017 at 21:52

31 votes

Accepted

Find $x,y,z>0$ such that $x+y+z=1$ and $x^2+y^2+z^2$ is minimal

calculus multivariable-calculus optimization nonlinear-optimization maxima-minima

answered Dec 12, 2015 at 13:54

29 votes

Accepted

Can multiples of two reals stay separated?

irrational-numbers irrationality-measure

answered Jun 8, 2018 at 9:00

29 votes

An exotic sequence

real-analysis sequences-and-series

answered Mar 10, 2014 at 5:38

27 votes

Summation of an infinite Exponential series

limits summation exponential-function

answered Jun 8, 2018 at 7:57

27 votes

Accepted

How to evaluate $\int_{0}^{+\infty}\exp(-ax^2-\frac b{x^2})\,dx$ for $a,b>0$

calculus integration definite-integrals improper-integrals

answered Sep 17, 2013 at 5:38

26 votes

Accepted

How to compute $\int_{-\infty}^\infty\exp\left(-\frac{(x^2-13x-1)^2}{611x^2}\right)\ dx$

calculus integration definite-integrals improper-integrals closed-form

answered Sep 29, 2014 at 10:16

26 votes

Accepted

How prove this integral $ \int\limits_0^1 \int\limits_0^1 \ln\Gamma(x+y^3)\,dx\,dy =-\frac 7 {16}+\frac{1}{2}\ln 2\pi$

calculus integration multivariable-calculus definite-integrals gamma-function

answered Sep 24, 2013 at 5:05

26 votes

Accepted

How would you find the exact roots of $y=x^3+x^2-2x-1$?

algebra-precalculus trigonometry polynomials roots cubics

answered Jun 11, 2016 at 19:33

25 votes

Integral $\int_0^1 \log \frac{1+ax}{1-ax}\frac{dx}{x\sqrt{1-x^2}}=\pi\arcsin a$

calculus real-analysis integration complex-analysis definite-integrals

answered May 15, 2014 at 5:58

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2,087 Answers

Conjecture $\int_0^1\frac{\mathrm dx}{\sqrt{1-x}\ \sqrt[4]x\ \sqrt[4]{2-x\,\sqrt3}}\stackrel?=\frac{2\,\sqrt2}{3\,\sqrt[8]3}\pi$

Is $0.1010010001000010000010000001 \ldots$ transcendental?

Integral $\int_0^1\frac{\ln\left(x+\sqrt2\right)}{\sqrt{2-x}\,\sqrt{1-x}\,\sqrt{\vphantom{1}x}}\mathrm dx$

What are some examples of mathematics that had unintended useful applications much later?

Does non-symmetric positive definite matrix have positive eigenvalues?

Conjecture $\int_0^1\frac{dx}{\sqrt[3]x\,\sqrt[6]{1-x}\,\sqrt{1-x\left(\sqrt{6}\sqrt{12+7\sqrt3}-3\sqrt3-6\right)^2}}=\frac\pi9(3+\sqrt2\sqrt[4]{27})$

Find $xy+yz+zx$ given systems of three homogenous quadratic equations for $x, y, z$

Why does Wolframalpha think that this sum converges?

How does one [easily] calculate $\sum\limits_{n=1}^\infty\frac{\mathrm{pop}(n)}{n(n+1)}$?

Prove that there exists a triangle which can be cut into 2005 congruent triangles.

Is there a fast way to prove a tridiagonal matrix is positive definite?

Evaluating $\int_{0}^{1}\cdots\int_{0}^{1}\bigl\{\frac{1}{x_{1}\cdots x_{n}}\bigr\}^{2}\:\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}$

A strange integral: $\int_{-\infty}^{+\infty} {dx \over 1 + \left(x + \tan x\right)^2} = \pi.$

Conflicting limit answers using calculator and wolfram alpha

Can you complete the expression $2 \underline{ }\, \underline{ }\, \underline{ } \,\underline{ } 5 = 2015$?

Is Tolkien's Middle Earth flat?

Find large power of a non-diagonalisable matrix

Geometry problem involving infinite number of circles

How prove this matrix $\det (A)=\left(\frac{1}{\ln{(a_{i}+a_{j})}}\right)_{n\times n}\neq 0$

A very different property of primitive Pythagorean triplets: Can number be in more than two of them?

What's the limit of $\sqrt{2 + \sqrt{2-\sqrt{2+\sqrt{2-\sqrt{2-\sqrt{2 + ...}}}}}} $?

Find $x,y,z>0$ such that $x+y+z=1$ and $x^2+y^2+z^2$ is minimal

Can multiples of two reals stay separated?

An exotic sequence

Summation of an infinite Exponential series

How to evaluate $\int_{0}^{+\infty}\exp(-ax^2-\frac b{x^2})\,dx$ for $a,b>0$

How to compute $\int_{-\infty}^\infty\exp\left(-\frac{(x^2-13x-1)^2}{611x^2}\right)\ dx$

How prove this integral $ \int\limits_0^1 \int\limits_0^1 \ln\Gamma(x+y^3)\,dx\,dy =-\frac 7 {16}+\frac{1}{2}\ln 2\pi$

How would you find the exact roots of $y=x^3+x^2-2x-1$?

Integral $\int_0^1 \log \frac{1+ax}{1-ax}\frac{dx}{x\sqrt{1-x^2}}=\pi\arcsin a$