User jack

40 votes

5 answers

3k views

The sequence $a_{n+1}=\left\lceil \frac{-1+\sqrt{5}}{2}a_{n}-a_{n-1} \right\rceil$ is periodic

asked Jan 30, 2016 at 21:07

30 votes

3 answers

2k views

All polynomials are the sum of three others, each of which has only real roots

asked Mar 10, 2018 at 10:32

29 votes

6 answers

2k views

$m$-fold composite $p^{(m)}(x) \in \mathbb{Z}[x]$ implies $p(x) \in \mathbb{Z}[x]$

asked Dec 19, 2020 at 13:21

24 votes

2 answers

1k views

If $x_{n+1}= \frac{nx_{n}^2+1}{n+1}$ then $x_{n}=1$

asked Aug 12, 2017 at 14:28

20 votes

1 answer

754 views

Minimum value of $|p(1)|^2+|p(2)|^2 +...+ |p(n+3)|^2$ over all monic polynomials $p$

asked Jul 4, 2015 at 16:18

20 votes

2 answers

1k views

Find $Y\in\operatorname{GL}_n(\mathbb{Z})$ such that all eigenvalues of $YX$ are nonnegative

asked Apr 9, 2022 at 13:31

18 votes

3 answers

1k views

Show that $(\sum_{k=1}^{n}x_{k}\cos{k})^2+(\sum_{k=1}^{n}x_{k}\sin{k})^2\le (2+\frac{n}{4})\sum_{k=1}^{n}x^2_{k}$

asked Mar 7, 2020 at 15:42

10 votes

0 answers

213 views

Permutation of positive integers

asked Oct 26 at 13:39

10 votes

1 answer

345 views

Finding $q(x)$ such that $p(q(x))$ is reducible over $\mathbb{Q}[x]$

asked Apr 18, 2020 at 14:16

10 votes

2 answers

371 views

Divisibility condition implies $a_1=\dotsb=a_k$?

asked Dec 12, 2020 at 13:18

9 votes

2 answers

316 views

Dividing the edges and diagonals of a polygon among disjoint sub-polygons

asked Jul 18, 2015 at 15:39

7 votes

3 answers

957 views

Vector of integers such that almost all dot products are positive

asked Aug 24 at 14:44

6 votes

0 answers

138 views

Counting $K_4$ on two graphs sharing the same vertices

asked Aug 20, 2016 at 14:29

6 votes

0 answers

474 views

(In)finitely many natural numbers are not the sum or difference of two perfect powers

asked Sep 3, 2016 at 14:36

6 votes

2 answers

364 views

Triangles whose vertices and center have all the same color

asked Jul 4, 2017 at 14:22

6 votes

1 answer

441 views

Minimum number of operations necessary to arrive at any configuration

asked Apr 14, 2018 at 10:05

6 votes

1 answer

312 views

$p_n(x,y)=\sum_{i=0}^{n-1}x^{n-1-i}y^{i}$ is always an integer

asked Nov 17, 2018 at 15:37

5 votes

2 answers

307 views

Tiling a Jordan polygon

asked Aug 14, 2021 at 13:49

4 votes

1 answer

200 views

$2$-adic order bound for $P(x)$

asked Apr 3, 2021 at 13:27

4 votes

1 answer

271 views

$n$ variables Diophantine

asked Jan 30, 2021 at 14:26

4 votes

2 answers

229 views

Colouring Positive Integers

asked Mar 3, 2018 at 9:59

4 votes

2 answers

256 views

Sets $X,Y \subset [0,1]$, stronger than being measure $0$, such that $X+Y = [0,2]$

asked Jun 27, 2015 at 15:28

3 votes

0 answers

198 views

On the equation $x^3+y^3+z^3-2xyz=N$

asked Mar 24, 2018 at 14:07

3 votes

0 answers

133 views

Number of divisors of $2^k+1$

asked Feb 13, 2021 at 13:49

3 votes

1 answer

195 views

Are there infinitey many $n$ dividing $\epsilon_11^n + \epsilon_22^n + \cdots + \epsilon_kk^n$?

asked Feb 20, 2021 at 13:38

3 votes

0 answers

134 views

Permutation of a sequence, such that $y_i+y_{i+1}$ are all distinct

asked Aug 3, 2019 at 14:49

3 votes

0 answers

276 views

Infinitely many $n$ such that $\gcd(\lfloor n\sqrt{2}\rfloor, \lfloor n\sqrt{3}\rfloor)=m$

asked Jun 28, 2020 at 9:26

2 votes

1 answer

212 views

Covering the surface below a convex function

asked Oct 16, 2021 at 13:15

2 votes

0 answers

106 views

Decomposing a planar graph

asked Nov 6, 2021 at 12:48

2 votes

1 answer

192 views

Every element of $A$ and $B$ differ in at least $k$ positions

asked Apr 11, 2020 at 15:55

Stack Exchange Network

34 Questions

The sequence $a_{n+1}=\left\lceil \frac{-1+\sqrt{5}}{2}a_{n}-a_{n-1} \right\rceil$ is periodic

All polynomials are the sum of three others, each of which has only real roots

$m$-fold composite $p^{(m)}(x) \in \mathbb{Z}[x]$ implies $p(x) \in \mathbb{Z}[x]$

If $x_{n+1}= \frac{nx_{n}^2+1}{n+1}$ then $x_{n}=1$

Minimum value of $|p(1)|^2+|p(2)|^2 +...+ |p(n+3)|^2$ over all monic polynomials $p$

Find $Y\in\operatorname{GL}_n(\mathbb{Z})$ such that all eigenvalues of $YX$ are nonnegative

Show that $(\sum_{k=1}^{n}x_{k}\cos{k})^2+(\sum_{k=1}^{n}x_{k}\sin{k})^2\le (2+\frac{n}{4})\sum_{k=1}^{n}x^2_{k}$

Permutation of positive integers

Finding $q(x)$ such that $p(q(x))$ is reducible over $\mathbb{Q}[x]$

Divisibility condition implies $a_1=\dotsb=a_k$?

Dividing the edges and diagonals of a polygon among disjoint sub-polygons

Vector of integers such that almost all dot products are positive

Counting $K_4$ on two graphs sharing the same vertices

(In)finitely many natural numbers are not the sum or difference of two perfect powers

Triangles whose vertices and center have all the same color

Minimum number of operations necessary to arrive at any configuration

$p_n(x,y)=\sum_{i=0}^{n-1}x^{n-1-i}y^{i}$ is always an integer

Tiling a Jordan polygon

$2$-adic order bound for $P(x)$

$n$ variables Diophantine

Colouring Positive Integers

Sets $X,Y \subset [0,1]$, stronger than being measure $0$, such that $X+Y = [0,2]$

On the equation $x^3+y^3+z^3-2xyz=N$

Number of divisors of $2^k+1$

Are there infinitey many $n$ dividing $\epsilon_11^n + \epsilon_22^n + \cdots + \epsilon_kk^n$?

Permutation of a sequence, such that $y_i+y_{i+1}$ are all distinct

Infinitely many $n$ such that $\gcd(\lfloor n\sqrt{2}\rfloor, \lfloor n\sqrt{3}\rfloor)=m$

Covering the surface below a convex function

Decomposing a planar graph

Every element of $A$ and $B$ differ in at least $k$ positions