User TheSimpliFire

104 votes

1 answer

3k views

Can we remove any prime number with this strange process?

asked Jul 8, 2019 at 9:20

46 votes

1 answer

1k views

How to evaluate double limit of multifactorial $\lim\limits_{k\to\infty}\lim\limits_{n\to 0} \sqrt[n]{n\underbrace{!!!!\cdots!}_{k\,\text{times}}}$

real-analysis limits factorial closed-form gamma-function

asked Feb 18, 2021 at 10:31

44 votes

4 answers

1k views

Is each of $\int_0^\infty\frac{dx}{x^x},\int_0^\infty\frac{dx}{x^{x^{x^x}}},\int_0^\infty\frac{dx}{x^{x^{x^{x^{x^x}}}}},\cdots$ less than $2$?

integration special-functions power-towers

asked Feb 15 at 17:52

35 votes

2 answers

1k views

Is $100$ the only square number of the form $a^b+b^a$?

number-theory modular-arithmetic square-numbers conjectures pythagorean-triples

asked Jul 14, 2019 at 10:19

33 votes

1 answer

1k views

Show that the maximum value of this nested radical is $\phi-1$

functions recursion maxima-minima nested-radicals golden-ratio

asked Jan 5, 2019 at 16:41

29 votes

3 answers

1k views

Sum of digits of $a^b$ equals $ab$

number-theory decimal-expansion conjectures

asked Feb 24, 2019 at 17:10

27 votes

4 answers

1k views

Show that $\left(\frac{x_1^{x_2}}{x_2}\right)^p+\left(\frac{x_2^{x_3}}{x_3}\right)^p+\cdots+\left(\frac{x_n^{x_1}}{x_1}\right)^p\ge n$ for any $p\ge1$

real-analysis inequality

asked Oct 16, 2021 at 10:32

26 votes

1 answer

714 views

Are these continued fractions of integrals known?

integration reference-request maxima-minima continued-fractions

asked Jul 6, 2019 at 19:55

24 votes

7 answers

996 views

Prove that $\int_0^\infty\frac1{x^x}\, dx<2$

integration definite-integrals summation special-functions

asked Dec 30, 2017 at 16:20

24 votes

2 answers

722 views

Integral $\int_0^\infty\frac{\log(1+\cos x)}{1+e^x}\,dx=0$

real-analysis integration complex-analysis definite-integrals

asked May 4, 2021 at 15:10

20 votes

1 answer

521 views

Interesting patterns in $f(k,n)=k\pi-\sum\limits_{x=1}^n\tan^{-1}\left(\frac1{\sqrt[k]x}\right)$

sequences-and-series trigonometry optimization recreational-mathematics

asked Aug 3, 2019 at 10:22

20 votes

1 answer

635 views

On the integral $\int_{-\pi/2}^{\pi/2}\sin(x/\sin(x/\sin(x/\sin\cdots)))\,dx$

integration limits convergence-divergence regression

asked Oct 7, 2018 at 12:23

19 votes

1 answer

533 views

On the integral $\int_0^\pi\sin(x+\sin(x+\sin(x+\cdots)))\,dx$

integration limits convergence-divergence

asked Oct 6, 2018 at 10:03

18 votes

0 answers

343 views

Conjecture: No positive integer can be written as $a^b+b^a$ in more than one way

number-theory modular-arithmetic diophantine-equations conjectures

asked Jul 7, 2019 at 19:39

18 votes

1 answer

607 views

On the proximity of $a\sqrt b+b\sqrt a$ to an integer

number-theory irrationality-measure

asked Jun 27, 2020 at 15:37

17 votes

2 answers

469 views

Evaluating $1-\frac12-\frac13+\frac14+\frac15+\frac16-\cdots$

sequences-and-series convergence-divergence closed-form harmonic-numbers

asked Dec 10, 2018 at 20:41

13 votes

2 answers

296 views

Inequality $(1-e^x)\ln(1-xe^{-x})\leq x^2$

calculus inequality

asked Jul 19 at 23:23

12 votes

8 answers

555 views

Prove that $\int_{-\infty}^\infty\frac{e^{-tx^2}}{\cosh\pi x}\,dx\ge\frac{e^{t/2}}{t+1}$ for all $t\in[0,1]$

integration inequality

asked Jun 9 at 21:37

12 votes

1 answer

478 views

On the integral $\int_0^\pi\sin(x\sin(x\sin(x\cdots)))\,dx$

integration limits convergence-divergence

asked Oct 7, 2018 at 9:19

12 votes

2 answers

428 views

An interesting integral $\cos(x)\cos(x^2)\cos(x^3)...$

trigonometric-integrals upper-lower-bounds

asked Feb 16, 2018 at 16:33

11 votes

1 answer

473 views

What is $\underbrace{2018^{2018^{2018^{\mathstrut^{.^{.^{.^{2018}}}}}}}}_{p\,\text{times}}\pmod p$ where $p$ is an odd prime?

elementary-number-theory prime-numbers modular-arithmetic tetration big-numbers

asked Feb 10, 2018 at 11:40

11 votes

3 answers

1k views

Finding the turning points of $f(x)=\left(x-a+\frac1{ax}\right)^a-\left(\frac1x-\frac1a+ax\right)^x$

functions derivatives implicit-differentiation stationary-point

asked Dec 2, 2017 at 21:40

10 votes

1 answer

483 views

Area of a mushroom-shaped curve

integration definite-integrals recreational-mathematics

asked Jul 17, 2018 at 19:54

10 votes

3 answers

944 views

Some interesting observations on a sum of reciprocals

polynomials convergence-divergence roots

asked Aug 7, 2018 at 14:15

10 votes

0 answers

123 views

Conjecture: Is the identity $2^5-5^2=2+5$ unique? [duplicate]

elementary-number-theory diophantine-equations conjectures

asked Jul 11, 2019 at 16:09

9 votes

2 answers

2k views

Functions where the sum of its partial derivatives is zero

partial-differential-equations

asked Oct 16, 2019 at 9:18

9 votes

1 answer

251 views

A functional analogue to a finite geometric series

real-analysis functional-equations

asked Oct 4, 2020 at 14:32

8 votes

0 answers

181 views

Probability that $\int_0^tX_s\,dW_s$ lies within $1/t$ of $X_t$

probability stochastic-processes asymptotics stochastic-calculus brownian-motion

asked May 22, 2021 at 14:47

8 votes

0 answers

171 views

Is there any example of a real-analytic approach to evaluate a definite integral (with an elementary integrand) whose value involves Lambert W?

real-analysis integration definite-integrals lambert-w elementary-functions

asked Jul 28 at 9:40

8 votes

0 answers

90 views

Are there infinitely many solutions such that the digit sum of a prime power is a smaller power of the same prime?

number-theory prime-numbers decimal-expansion conjectures

asked Apr 22, 2019 at 7:20

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100 Questions

Can we remove any prime number with this strange process?

How to evaluate double limit of multifactorial $\lim\limits_{k\to\infty}\lim\limits_{n\to 0} \sqrt[n]{n\underbrace{!!!!\cdots!}_{k\,\text{times}}}$

Is each of $\int_0^\infty\frac{dx}{x^x},\int_0^\infty\frac{dx}{x^{x^{x^x}}},\int_0^\infty\frac{dx}{x^{x^{x^{x^{x^x}}}}},\cdots$ less than $2$?

Is $100$ the only square number of the form $a^b+b^a$?

Show that the maximum value of this nested radical is $\phi-1$

Sum of digits of $a^b$ equals $ab$

Show that $\left(\frac{x_1^{x_2}}{x_2}\right)^p+\left(\frac{x_2^{x_3}}{x_3}\right)^p+\cdots+\left(\frac{x_n^{x_1}}{x_1}\right)^p\ge n$ for any $p\ge1$

Are these continued fractions of integrals known?

Prove that $\int_0^\infty\frac1{x^x}\, dx<2$

Integral $\int_0^\infty\frac{\log(1+\cos x)}{1+e^x}\,dx=0$

Interesting patterns in $f(k,n)=k\pi-\sum\limits_{x=1}^n\tan^{-1}\left(\frac1{\sqrt[k]x}\right)$

On the integral $\int_{-\pi/2}^{\pi/2}\sin(x/\sin(x/\sin(x/\sin\cdots)))\,dx$

On the integral $\int_0^\pi\sin(x+\sin(x+\sin(x+\cdots)))\,dx$

Conjecture: No positive integer can be written as $a^b+b^a$ in more than one way

On the proximity of $a\sqrt b+b\sqrt a$ to an integer

Evaluating $1-\frac12-\frac13+\frac14+\frac15+\frac16-\cdots$

Inequality $(1-e^x)\ln(1-xe^{-x})\leq x^2$

Prove that $\int_{-\infty}^\infty\frac{e^{-tx^2}}{\cosh\pi x}\,dx\ge\frac{e^{t/2}}{t+1}$ for all $t\in[0,1]$

On the integral $\int_0^\pi\sin(x\sin(x\sin(x\cdots)))\,dx$

An interesting integral $\cos(x)\cos(x^2)\cos(x^3)...$

What is $\underbrace{2018^{2018^{2018^{\mathstrut^{.^{.^{.^{2018}}}}}}}}_{p\,\text{times}}\pmod p$ where $p$ is an odd prime?

Finding the turning points of $f(x)=\left(x-a+\frac1{ax}\right)^a-\left(\frac1x-\frac1a+ax\right)^x$

Area of a mushroom-shaped curve

Some interesting observations on a sum of reciprocals

Conjecture: Is the identity $2^5-5^2=2+5$ unique? [duplicate]

Functions where the sum of its partial derivatives is zero

A functional analogue to a finite geometric series

Probability that $\int_0^tX_s\,dW_s$ lies within $1/t$ of $X_t$

Is there any example of a real-analytic approach to evaluate a definite integral (with an elementary integrand) whose value involves Lambert W?

Are there infinitely many solutions such that the digit sum of a prime power is a smaller power of the same prime?