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ə̷̶̸͇̘̜́̍͗̂̄︣͟
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108 votes
1 answer
3k views

Can we remove any prime number with this strange process?

48 votes
1 answer
1k views

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

46 votes
4 answers
2k 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 $2$?

37 votes
2 answers
1k views

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

36 votes
2 answers
1k views

Are these continued fractions of integrals known?

33 votes
1 answer
1k views

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

33 votes
3 answers
1k views

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

29 votes
8 answers
2k views

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

28 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 $p\ge1$

24 votes
2 answers
821 views

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

20 votes
0 answers
400 views

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

20 votes
1 answer
532 views

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

20 votes
1 answer
702 views

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

19 votes
1 answer
596 views

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

19 votes
1 answer
620 views

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

17 votes
1 answer
475 views

Signs of entries of Kravchuk matrices asymptotically produce a large circular region with hyperbolic sinks. Why?

17 votes
2 answers
506 views

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

13 votes
8 answers
606 views

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

13 votes
2 answers
322 views

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

12 votes
1 answer
554 views

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

12 votes
2 answers
514 views

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

11 votes
1 answer
496 views

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

11 votes
3 answers
1k views

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

10 votes
1 answer
771 views

Area of a mushroom-shaped curve

10 votes
3 answers
1k views

Some interesting observations on a sum of reciprocals

10 votes
0 answers
371 views

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

10 votes
0 answers
129 views

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

9 votes
2 answers
2k views

Functions where the sum of its partial derivatives is zero

9 votes
1 answer
261 views

A functional analogue to a finite geometric series

8 votes
0 answers
188 views

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