User John M. Campbell

44 votes

2 answers

3k views

What is $\, _4F_3\left(1,1,1,\frac{3}{2};\frac{5}{2},\frac{5}{2},\frac{5}{2};1\right)$?

asked Feb 1, 2017 at 1:16

23 votes

2 answers

1k views

What is $\int_0^1 \ln (1-x) \ln \left(\ln \left(\frac{1}{x}\right)\right) \, dx$?

asked Apr 8, 2016 at 1:27

20 votes

3 answers

4k views

What is $\int_0^1 \frac{\log \left(1-x^2\right) \sin ^{-1}(x)^2}{x^2} \, dx$?

asked Aug 23, 2016 at 3:26

19 votes

1 answer

676 views

What is the number of $n \times n$ binary matrices $A$ such that $\det(A) = \text{perm}(A)$?

asked Mar 21, 2016 at 16:21

17 votes

1 answer

312 views

Is it true that $\mathbb{C}(x) \equiv \mathbb{C}(x, y)$?

asked Mar 9, 2016 at 10:19

17 votes

1 answer

410 views

Is there a closed-form expression for $\sum_{k=1}^{n}\lfloor k^{q} \rfloor$ for $q \in \mathbb{Q}_{> 0}$?

asked Mar 11, 2016 at 13:32

15 votes

1 answer

312 views

Examples of classes $\mathcal{C}$ of structures such that every finite group is isomorphic to the automorphism group of a structure in $\mathcal{C}$

asked Mar 25, 2016 at 19:53

14 votes

1 answer

932 views

Is the Fibonacci constant $0.11235813213455...$ a normal number?

asked Jun 30, 2016 at 18:55

12 votes

1 answer

457 views

A problem involving the product $\prod_{k=1}^{n} k^{\mu(k)}$, where $\mu$ denotes the Möbius function

asked Mar 12, 2016 at 20:22

10 votes

1 answer

546 views

What are some interesting counterexamples given by finite topological spaces?

asked Mar 14, 2016 at 17:40

10 votes

2 answers

551 views

Is there a simple way of proving that $\text{GL}_n(R) \not\cong \text{GL}_m(R)$?

asked Mar 7, 2016 at 7:06

9 votes

1 answer

194 views

Is it true that $\mathbb{F}_{1}^{\ast} \equiv \mathbb{F}_{2}^{\ast}$ implies $\mathbb{F}_{1} \equiv \mathbb{F}_{2}$?

asked Mar 8, 2016 at 16:34

8 votes

2 answers

475 views

What is the subword complexity function of this infinite word?

asked Apr 15, 2016 at 2:33

8 votes

1 answer

357 views

What is the inverse of $\left[ \sum_{k=1}^{j} \left\lfloor \frac{i}{k} \right\rfloor \right]_{n \times n}$?

asked Feb 25, 2018 at 23:43

6 votes

1 answer

513 views

What is $\int_{0}^{\infty}\frac{\cos^{n}x \sin x \ln x}{x} dx$?

asked Apr 5, 2016 at 20:42

5 votes

3 answers

207 views

What is $\prod _{j=1}^n \left(\sqrt{j}+1\right)$?

asked Jul 26, 2016 at 12:56

5 votes

1 answer

180 views

What is $\sum_{i=1}^{n}\frac{F_i}{i}$?

asked Jul 6, 2016 at 3:25

5 votes

0 answers

94 views

A conjecture concerning the irreducibility of characteristic polynomials of Arndt matrices

asked Mar 16, 2016 at 0:16

5 votes

1 answer

494 views

Given a field $\mathbb{F}$, what is $\text{Aut}(\mathbb{F}^{\ast})$?

asked Mar 16, 2016 at 16:59

4 votes

1 answer

331 views

What is $\sum_{i=0}^n \left\lfloor \sqrt{i}\right\rfloor \binom{n}{i}$?

asked Jun 28, 2016 at 17:55

4 votes

0 answers

221 views

How can we prove a closed form for $\frac{1}{8} \text{Li}_2\left(\frac{2+\sqrt{3}}{4} \right)+\text{Li}_2\left(2+\sqrt{3}\right)$?

asked Apr 24, 2022 at 16:41

2 votes

0 answers

87 views

An interesting connection between the Möbius function and the parity of the number of sublattices of index $n$ in generic $3$-dimensional lattice

asked Jul 18, 2016 at 2:27

2 votes

1 answer

581 views

Is there a simple way of proving that $\lfloor\sqrt{n}\rfloor+\lfloor\sqrt{4n+1}\rfloor = \lfloor\frac{3}{2} \lfloor \sqrt{4n+1} \rfloor\rfloor$?

asked Mar 22, 2016 at 18:48

2 votes

1 answer

126 views

What is $\left(\overline{\mathbb{C}(t)}\right)^{\times}$?

asked Mar 24, 2016 at 2:20

2 votes

1 answer

131 views

An example of a sentence $\sigma$ s.t. $\text{GL}_n(\mathbb{Q}(\sqrt{3})) \models \sigma$ and $\text{GL}_n(\mathbb{Q}(\sqrt{2})) \not\models \sigma$

asked Mar 10, 2016 at 13:21

1 vote

1 answer

528 views

What is the average determinant of a matrix in $M_{2}(\mathbb{Z}/n\mathbb{Z})$?

asked Apr 5, 2016 at 1:35

1 vote

0 answers

39 views

Let $f(x) = x^2$ if $x\leq 8$, and let $f(x) = \sqrt[3]{x}$ otherwise. What is $\lim_{i \to \infty} \frac{f^{(i)}(2)}{i}$?

asked Jul 12, 2016 at 0:47

1 vote

3 answers

369 views

Without using Cauchy's theorems, how can we prove that $F'(z) = \frac{1}{z}$ implies that $F$ isn't holomorphic on a given annulus?

asked May 7, 2017 at 16:00

1 vote

2 answers

248 views

What is $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n^{3} + 1}}$?

asked Aug 10, 2017 at 1:21

0 votes

0 answers

41 views

What is $Z(\mathbb{Z}_{p} \rtimes_{\phi} \mathbb{Z}_{q})$ if $\phi$ is non-trivial and $p$ and $q$ are primes such that $q \equiv 1(\text{mod} \ p)$?

asked Oct 28, 2016 at 5:34

Stack Exchange Network

31 Questions

What is $\, _4F_3\left(1,1,1,\frac{3}{2};\frac{5}{2},\frac{5}{2},\frac{5}{2};1\right)$?

What is $\int_0^1 \ln (1-x) \ln \left(\ln \left(\frac{1}{x}\right)\right) \, dx$?

What is $\int_0^1 \frac{\log \left(1-x^2\right) \sin ^{-1}(x)^2}{x^2} \, dx$?

What is the number of $n \times n$ binary matrices $A$ such that $\det(A) = \text{perm}(A)$?

Is it true that $\mathbb{C}(x) \equiv \mathbb{C}(x, y)$?

Is there a closed-form expression for $\sum_{k=1}^{n}\lfloor k^{q} \rfloor$ for $q \in \mathbb{Q}_{> 0}$?

Examples of classes $\mathcal{C}$ of structures such that every finite group is isomorphic to the automorphism group of a structure in $\mathcal{C}$

Is the Fibonacci constant $0.11235813213455...$ a normal number?

A problem involving the product $\prod_{k=1}^{n} k^{\mu(k)}$, where $\mu$ denotes the Möbius function

What are some interesting counterexamples given by finite topological spaces?

Is there a simple way of proving that $\text{GL}_n(R) \not\cong \text{GL}_m(R)$?

Is it true that $\mathbb{F}_{1}^{\ast} \equiv \mathbb{F}_{2}^{\ast}$ implies $\mathbb{F}_{1} \equiv \mathbb{F}_{2}$?

What is the subword complexity function of this infinite word?

What is the inverse of $\left[ \sum_{k=1}^{j} \left\lfloor \frac{i}{k} \right\rfloor \right]_{n \times n}$?

What is $\int_{0}^{\infty}\frac{\cos^{n}x \sin x \ln x}{x} dx$?

What is $\prod _{j=1}^n \left(\sqrt{j}+1\right)$?

What is $\sum_{i=1}^{n}\frac{F_i}{i}$?

A conjecture concerning the irreducibility of characteristic polynomials of Arndt matrices

Given a field $\mathbb{F}$, what is $\text{Aut}(\mathbb{F}^{\ast})$?

What is $\sum_{i=0}^n \left\lfloor \sqrt{i}\right\rfloor \binom{n}{i}$?

How can we prove a closed form for $\frac{1}{8} \text{Li}_2\left(\frac{2+\sqrt{3}}{4} \right)+\text{Li}_2\left(2+\sqrt{3}\right)$?

An interesting connection between the Möbius function and the parity of the number of sublattices of index $n$ in generic $3$-dimensional lattice

Is there a simple way of proving that $\lfloor\sqrt{n}\rfloor+\lfloor\sqrt{4n+1}\rfloor = \lfloor\frac{3}{2} \lfloor \sqrt{4n+1} \rfloor\rfloor$?

What is $\left(\overline{\mathbb{C}(t)}\right)^{\times}$?

An example of a sentence $\sigma$ s.t. $\text{GL}_n(\mathbb{Q}(\sqrt{3})) \models \sigma$ and $\text{GL}_n(\mathbb{Q}(\sqrt{2})) \not\models \sigma$

What is the average determinant of a matrix in $M_{2}(\mathbb{Z}/n\mathbb{Z})$?

Let $f(x) = x^2$ if $x\leq 8$, and let $f(x) = \sqrt[3]{x}$ otherwise. What is $\lim_{i \to \infty} \frac{f^{(i)}(2)}{i}$?

Without using Cauchy's theorems, how can we prove that $F'(z) = \frac{1}{z}$ implies that $F$ isn't holomorphic on a given annulus?

What is $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n^{3} + 1}}$?

What is $Z(\mathbb{Z}_{p} \rtimes_{\phi} \mathbb{Z}_{q})$ if $\phi$ is non-trivial and $p$ and $q$ are primes such that $q \equiv 1(\text{mod} \ p)$?