User mathlove

22 votes

1 answer

10k views

Prove that ${\sqrt2}^{\sqrt2}$ is an irrational number without using a theorem

asked Jul 31, 2013 at 14:35

38 votes

3 answers

5k views

Arranging numbers from $1$ to $n$ such that the sum of every two adjacent numbers is a perfect power

asked Mar 11, 2015 at 11:55

33 votes

3 answers

4k views

Can we simplify $\int_{0}^{\infty}\frac{{\sin}^px}{x^q}dx$?

asked Dec 1, 2013 at 14:57

24 votes

4 answers

2k views

Does this sequence always give an integer?

asked Sep 30, 2013 at 14:29

24 votes

4 answers

2k views

Letting $S(m)$ be the digit sum of $m$, then $\lim_{n\to\infty}S(3^n)=\infty$?

asked Sep 26, 2013 at 14:54

7 votes

1 answer

2k views

The number of distinct prime factors of $n\in\mathbb N$

asked Nov 23, 2013 at 15:15

13 votes

2 answers

2k views

Proving $\sum_{k=0}^{2m}(-1)^k{\binom{2m}{k}}^3=(-1)^m\binom{2m}{m}\binom{3m}{m}$

asked Sep 27, 2013 at 15:33

34 votes

4 answers

2k views

About the ratio of the areas of a convex pentagon and the inner pentagon made by the five diagonals

asked Oct 1, 2013 at 8:29

5 votes

2 answers

2k views

Solving $x^k+(x+1)^k+(x+2)^k+\cdots+(x+k-1)^k=(x+k)^k$ for $k\in\mathbb N$

asked Sep 12, 2013 at 15:11

10 votes

3 answers

2k views

On maximal regular polyhedra inscribed in a regular polyhedron

asked Jul 31, 2013 at 6:36

17 votes

5 answers

1k views

Proving that every term of the sequence is an integer

asked May 29, 2013 at 7:06

5 votes

3 answers

1k views

Finding an invisible circle by drawing another line

asked Aug 26, 2013 at 9:55

14 votes

3 answers

1k views

What is the max of $n$ such that $\sum_{i=1}^n\frac{1}{a_i}=1$ where $2\le a_1\lt a_2\lt \cdots\lt a_n\le m$?

asked Sep 16, 2013 at 14:41

5 votes

2 answers

1k views

About two 'negative' continued fractions whose sum equals $1$

asked Oct 16, 2013 at 15:15

12 votes

3 answers

1k views

A special tessellation

asked Nov 13, 2013 at 11:31

17 votes

1 answer

919 views

If $\left(1^a+2^a+\cdots+n^{a}\right)^b=1^c+2^c+\cdots+n^c$ for some $n$, then $(a,b,c)=(1,2,3)$?

asked Oct 7, 2013 at 14:27

20 votes

1 answer

876 views

Tetrahedra passing through a hole

asked Aug 7, 2013 at 6:29

2 votes

2 answers

770 views

Examples that the Fermat-Catalan conjecture does not cover

asked Feb 22, 2015 at 18:13

5 votes

0 answers

767 views

Conjectures on perfect squares

asked Feb 15, 2015 at 17:34

11 votes

0 answers

723 views

Making a convex polyhedron with two sheets of paper

asked Feb 24, 2014 at 11:31

2 votes

2 answers

658 views

Numbers $n$ such that the sum of the divisors of $n$ is a nontrivial power

asked Mar 24, 2015 at 10:50

2 votes

2 answers

644 views

Defining $\{a_i\}$ as $(1+x+⋯+x^k)^n =\sum_{i=0}^{kn}a_ix^i$, then is the 'special' difference-sequence $\{d^Na_i\}$ a unimodal sequence?

asked Sep 19, 2013 at 7:13

13 votes

0 answers

636 views

Representing every positive rational number in the form of $(a^n+b^n)/(c^n+d^n)$

asked Mar 21, 2015 at 9:29

4 votes

1 answer

564 views

About the inscribed sphere and the exspheres of a $n$-dimensional simplex

asked Nov 24, 2013 at 15:32

0 votes

1 answer

557 views

Permutations of letters under some conditions

asked Nov 29, 2013 at 14:53

3 votes

3 answers

556 views

An inequality about the sum of some unit fractions with a property

asked Oct 18, 2013 at 15:10

9 votes

1 answer

544 views

What is the shape of the $n$-gon which gives the maximum of a function?

asked Jul 14, 2013 at 14:29

13 votes

0 answers

543 views

If $\beta=0.{a_1}^{k}{a_2}^{k}{a_3}^{k}\cdots\in\mathbb Q$, then $\alpha=0.a_1a_2a_3\cdots\in\mathbb Q$?

asked Dec 7, 2013 at 15:58

5 votes

1 answer

527 views

Permutation polynomials mod $p$ of the form $(x+1)^n-x^n$

asked Mar 10, 2015 at 11:23

8 votes

0 answers

509 views

Proving a proposition which leads the irrationality of $\frac{\zeta(5)}{\zeta(2)\zeta(3)}$

asked Nov 7, 2013 at 15:07

Stack Exchange Network

45 Questions

Prove that ${\sqrt2}^{\sqrt2}$ is an irrational number without using a theorem

Arranging numbers from $1$ to $n$ such that the sum of every two adjacent numbers is a perfect power

Can we simplify $\int_{0}^{\infty}\frac{{\sin}^px}{x^q}dx$?

Does this sequence always give an integer?

Letting $S(m)$ be the digit sum of $m$, then $\lim_{n\to\infty}S(3^n)=\infty$?

The number of distinct prime factors of $n\in\mathbb N$

Proving $\sum_{k=0}^{2m}(-1)^k{\binom{2m}{k}}^3=(-1)^m\binom{2m}{m}\binom{3m}{m}$

About the ratio of the areas of a convex pentagon and the inner pentagon made by the five diagonals

Solving $x^k+(x+1)^k+(x+2)^k+\cdots+(x+k-1)^k=(x+k)^k$ for $k\in\mathbb N$

On maximal regular polyhedra inscribed in a regular polyhedron

Proving that every term of the sequence is an integer

Finding an invisible circle by drawing another line

What is the max of $n$ such that $\sum_{i=1}^n\frac{1}{a_i}=1$ where $2\le a_1\lt a_2\lt \cdots\lt a_n\le m$?

About two 'negative' continued fractions whose sum equals $1$

A special tessellation

If $\left(1^a+2^a+\cdots+n^{a}\right)^b=1^c+2^c+\cdots+n^c$ for some $n$, then $(a,b,c)=(1,2,3)$?

Tetrahedra passing through a hole

Examples that the Fermat-Catalan conjecture does not cover

Conjectures on perfect squares

Making a convex polyhedron with two sheets of paper

Numbers $n$ such that the sum of the divisors of $n$ is a nontrivial power

Defining $\{a_i\}$ as $(1+x+⋯+x^k)^n =\sum_{i=0}^{kn}a_ix^i$, then is the 'special' difference-sequence $\{d^Na_i\}$ a unimodal sequence?

Representing every positive rational number in the form of $(a^n+b^n)/(c^n+d^n)$

About the inscribed sphere and the exspheres of a $n$-dimensional simplex

Permutations of letters under some conditions

An inequality about the sum of some unit fractions with a property

What is the shape of the $n$-gon which gives the maximum of a function?

If $\beta=0.{a_1}^{k}{a_2}^{k}{a_3}^{k}\cdots\in\mathbb Q$, then $\alpha=0.a_1a_2a_3\cdots\in\mathbb Q$?

Permutation polynomials mod $p$ of the form $(x+1)^n-x^n$

Proving a proposition which leads the irrationality of $\frac{\zeta(5)}{\zeta(2)\zeta(3)}$