Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
List of integers without any arithmetic progression of n terms Let's consider a positive integer $n$ and the list of the $n^2$ integers from $1$ to $n^2$. What is the minimum number $f(n)$ of integers to be cancelled in this list so that it is impossible to form any arithmetic progression of $n$ terms with the remaini... | Extending my comment to an answer for small $n$: a brute force program gives
$$
\begin{align}
f(1)&=1 &&\{1\} \\
f(2)&=3 &&\{1,2,3\} \\
f(3)&=4 &&\{3,4,5,7\} \\
f(4)&=6 &&\{3,4,5,6,10,13\} \\
f(5)&=7 &&\{3, 7, 9, 10, 11, 16, 21\} \\
f(6)&=9 &&\{5, 8, 12, 14, 15, 16, 21, 26, 31\} \\
f(7)&=11 &&\{3, 9, 11, 12, 13, 14, ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/181649",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 4,
"answer_id": 0
} |
From a (not positive definite) Gram matrix to a (Kac-Moody) Cartan matrix Suppose I am given a symmetric matrix $G_{ij}$ with $G_{ii} = 2$. Can I always find an invertible integer matrix $S$ such that $(S^T G S)_{ii}=2$ and $(S^T G S)_{ij} \leq 0$ for $i \neq j$? Is there a practical algorithm to do so?
If you'd like a... | Wondering how well this would work out in $4$ by $4,$ given that David says the individual steps in the "algorithm" preserve something important:
$$ G =
\left(
\begin{array}{cccc}
2 & a & b & c \\
a & 2 & d & e \\
b & d & 2 & f \\
c & e & f& 2
\end{array}
\right)
$$
$$ \det G = a^2 f^2 + b^2 e^2 + c^2 d^2 - 2 (... | {
"language": "en",
"url": "https://mathoverflow.net/questions/208562",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
} |
Finding matrices $A$ such that the entries of $A^n$ have specified signs What techniques are there for ensuring nonnegativity of various entries of matrix powers?
Specific Question: Consider a matrix $A\in SL_2(\mathbb R)$. Let $(A^n)_{i,j}$ denote the $(i,j)$ entry of the matrix power $A^n$. Under what conditions on $... | To follow up on Terry's comment (actually, I did the computation before seeing it, but whatever): in the parabolic case, where the matrix has the form
$$A = \begin{pmatrix} d & -b \\ -c & a\end{pmatrix} \begin{pmatrix} 1 & x \\ 0 & 1\end{pmatrix} \begin{pmatrix} a & b \\ c & d\end{pmatrix},$$
Then
$$A^n = \begin{pmatr... | {
"language": "en",
"url": "https://mathoverflow.net/questions/211025",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Difference between maxima of random variables Given four independent, identically distributed Gaussian random variables with zero mean and unit variance $x_1$, $x_2$, $y_1$, $y_2$, consider
\begin{equation}
u \equiv \max(x_1+C\, y_1, x_2+C \, y_2) - \max(x_1-C \, y_1, x_2-C \, y_2),
\end{equation}
where $C$ is a real ... | Using $\max(a,b) = \dfrac{a+b}{2} + \left| \dfrac{a-b}{2}\right|$, write $u = w_1 + |w_2| - |w_3|$ where
$$ \eqalign{ w_1 &= C (y_1 + y_2) \cr
w_2 &= \dfrac{1}{2} (x_2 - x_1 + C (y_2 - y_1))\cr
w_3 &= \dfrac{1}{2} (x_2 - x_1 - C (y_2 - y_1))\cr}$$
are jointly normal with mean $0$ and covaria... | {
"language": "en",
"url": "https://mathoverflow.net/questions/232406",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
The coefficient of a specific monomial of the following polynomial Let the real polynomial
$$f_{a,b,c}(x_1,x_2,x_3)=(x_1-x_2)^{2a+1}(x_2-x_3)^{2b+1}(x_3-x_1)^{2c+1},$$
where $a,b,c$ are nonnegative integers.
Let $m_{a,b,c}$ be the coefficient of the monomial $x_1^{a+c+1}x_2^{a+b+1}x_3^{b+c+1}$ in the expansion of $f_{... | The coefficient is always zero. I use the superb conventions of Concrete Mathematics: Sums are over all integers unless otherwise indicated, and $\binom{n}{k}$ is $0$ if $k<0$ or $>n \geq 0$.
Expanding by the binomial theorem,
$$f_{abc} = \sum_{i,j,k} (-1)^{i+j+k} \binom{2a+1}{i} \binom{2b+1}{j} \binom{2c+1}{k} x_1^{2... | {
"language": "en",
"url": "https://mathoverflow.net/questions/234564",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 3
} |
Eigenvectors of a matrix with entries involving combinatorics In the question Eigenvalues of a matrix with entries involving combinatorics No_way asked about eigenvectors of $n\times n$ matrix $M$ with entries \begin{eqnarray*}
M_{ij}=(-1)^{i+j}F(n, l, i, j),
\end{eqnarray*}
where $F(n,l,i,j)$ is the cardinality of th... | In fact for a fixed $n$, the matrices $M(l, n)$ for $l>0$ commute with each other and thus are simultaneously diagonalisable. For your second question, if $\{p_j(y)\}$ is a sequence of polynomials satisfying
\begin{eqnarray}
\left(\frac{t}{\sinh t}\right)^y=\sum_{j=0}^\infty p_j(y)t^{2j}.
\end{eqnarray}
then the $i$-t... | {
"language": "en",
"url": "https://mathoverflow.net/questions/255869",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
An extremal problem Let $f:[0,\pi]\to [0,\pi]$ be a diffeomorphism. How to prove that
$$P[f]:=\int_0^\pi \sin^2(x) \left(3+2 \frac{\sin^2(f(x))}{\sin^2 x}+(f'(x))^2\right)^2dx $$ attains its minimum for $f(x)\equiv x$?
| The following SymPy script
from sympy import *
x = Symbol('x')
f = Function('f')(x)
# define Lagrangian
L = (sin(x))**2 * (3 + 2*((sin(f))**2/(sin(x))**2) + (Derivative(f,x))**2)**2
# Euler-Lagrange equation
print euler_equations(L,f,x)
produces the output
[Eq(-4*(2*(Derivative(f(x), x)*Derivative(f(x), x, x) + 2*s... | {
"language": "en",
"url": "https://mathoverflow.net/questions/263201",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Is this BBP-type formula for $\ln 31$, $\ln 127$, and other Mersenne numbers also true? In this post, a binary BBP-type formula for Fermat numbers $F_m$ was discussed as (with a small tweak),
$$\ln(2^b+1) = \frac{b}{2^{a-1}}\sum_{n=0}^\infty\frac{1}{(2^a)^n}\left(\sum_{j=1}^{a-1}\frac{2^{a-1-j}}{an+j}+\sum_{k=1}^{a/b-... | I'll give a proof of the corrected version of your conjectured series expansion and then comment at the end about the limits of this method for finding base 2 BBP formulas for $\log n$.
Let's denote $\alpha_b=2^b-2-b\lfloor\frac{2^b-2}{b}\rfloor$, notice that $\alpha_b=0$ for all primes $b$ but it can be nonzero for ot... | {
"language": "en",
"url": "https://mathoverflow.net/questions/273894",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Inverse of special upper triangular matrix Consider the following $n \times n$ upper triangular matrix with a particularly nice structure:
\begin{equation}\mathbf{P} = \begin{pmatrix}
1 & \beta & \alpha+\beta & \dots & (n-3)\alpha + \beta & (n-2)\alpha + \beta\\
0 & 1 & \beta & \dots & (n-4)\alpha + \beta & (n-3)\alpha... | Let $A$ be the nilpotent matrix $$\begin{pmatrix}0 & 1 & 1 & \cdots & 1 \\ & 0 & 1 & \cdots & 1 \\ & & \cdots & \cdots & \cdots \\ & & & 0 & 1 \\ & & & & 0\end{pmatrix},$$ then the matrix $P$ is equal to $1 + \beta A + \alpha A^2$.
This gives the inverse: \begin{eqnarray*}P^{-1} & = & (1 + \beta A + \alpha A^2)^{-1} \\... | {
"language": "en",
"url": "https://mathoverflow.net/questions/279842",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 1
} |
A property of 47 with respect to partitions into five parts Is 47 the largest number which has a unique partition into five parts (15, 10, 10, 6, 6), no two of which are relatively prime?
| Yes. Suppose $n>47$.
If $2\mid n$, we can take $(n-8,2,2,2,2),(n-10,4,2,2,2)$, which are distinct partitions for $n\geq 14$.
If $3\mid n$, we can take $(n-12,3,3,3,3),(n-15,6,3,3,3)$, which are distinct partitions for $n\geq 21$.
If $n\equiv 1\pmod 6$, we can take $(n-37,15,10,6,6),(n-43,15,12,10,6)$, which are distinc... | {
"language": "en",
"url": "https://mathoverflow.net/questions/289084",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 1
} |
How to estimate a total variation distance? Let $X_1, \ldots, X_n$ be independent Bernoulli random variables. Then $Pr[X_i=1]=Pr[X_i=0]=1/2$. Let $X = (X_1, \ldots, X_n)$ and $v \in \{0,1\}^n$, $Y=v \cdot X$, $Z=Y-1$. Let
\begin{align}
\mu_1(x) = Pr[Y=x], \ \mu_2(x)=Pr[Z=x], \ x \in [n]=\{1, \ldots, n\}.
\end{align}
... | For the TV-distance, we have
\begin{align}
d_{TV}= \frac{1}{2} \sum_{x=0}^k |d_x|
\end{align}
(with the summation actually beginning at $x=0$), where
\begin{multline*}
d_x:=P(S_k=x) - P(S_k=x+1)=\frac1{2^k}\,\Big(\binom kx-\binom k{x+1}\Big) \\
=\frac1{2^k}\,\frac{k!}{(x+1)!(k-x)!}\,(2x-(k-1))
\end{multline*}
an... | {
"language": "en",
"url": "https://mathoverflow.net/questions/307502",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Solution for $Xa + X^Tb = c$ where $X^TX = I$? There are three known $n\times1$ vectors: $a, b, c$, along with one unknown $n\times n$ matrix: $X$. I am only interested in the $n={2,3}$ cases.
$X$ is $2\times 2$ or $3\times 3$ rotation matrix with an unusual domain specific constriant:
*
*$X^TX = XX^T = I$
*$Xa + ... | \begin{equation}
Xa = \begin{bmatrix} a_1x_{1,1} + a_2x_{1,2}\\ a_1x_{2,1} + a_2x_{2,2}\\
\end{bmatrix}
\end{equation}
and
\begin{equation}
X^Tb = \begin{bmatrix} b_1x_{1,1} + b_2x_{2,1}\\ b_1x_{1,2} + b_2x_{1,2}\\
\end{bmatrix}
\end{equation}
and
\begin{equation}
Xa+X^Tb = \begin{bmatrix}a_1x_{1,1} + a_2x_{1,2} ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/360344",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
} |
"Laurent phenomenon"? Define the recurrence
\begin{align*}
n(2n+x-3)u_n(x)
&=2(2n+x-2)(4n^2+4nx-8n-3x+3)u_{n-1}(x) \\
&-4(n+x-2)(2n-3)(2n+2x-3)(2n+x-1)u_{n-2}(x)
\end{align*}
with initial conditions $u_0(x)=0$ and $u_1(x)=x+1$.
The subject of "Laurent phenomenon" was motivated by Somos sequences. In the same spirit, I ... | In fact,
$$
u_n(x) =
{2}^{n-1}\prod _{k=0}^{n-1}(2x+2k+1)
-{2\,n-1\choose n-1}\prod _{k=0}^{n-1}(x+k) ,
\tag1$$
which is a polynomial with integer coefficients.
P.S. the proof rests on a routine verification that (1) satisfies the given recurrence relation and initial conditions.
| {
"language": "en",
"url": "https://mathoverflow.net/questions/378507",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 1,
"answer_id": 0
} |
Power of $2$ dividing a specialized Mittag-Leffler polynomial While studying the so-called Mittag-Leffler Polynomials, denoted $M_n(x)$, I was looking into the sequence $\frac1{n!}M_n(n)$ which takes the following form
$$a_n:=\sum_{k=1}^n\binom{n-1}{k-1}\binom{n}k2^k.$$
QUESTION 1. Let $\nu_2(m)$ denote the $2$-adic v... | I will confine myself to Question 1 since you mentioned that you know how to do Question 2. Also the case when $n$ is odd is easy, and let us restrict to $n$ being divisible exactly by $2^r$ with $r\ge 1$, and we need to show that the exact power of $2$ dividing $a_n$ is $3r$. Thus in what follows we may discard any ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/393532",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
When is $2^{2n}-2^n+1$ prime? Is it known when
$2^{2n}-2^n+1$
is prime?
It seems to be only when n is 1,2,4 or 32.
| Expanding on the remarks by Pace Nielsen (with an additional result that we need only consider $n>2$ a multiple of $4.$):
$x^n-1=\prod_{d \mid n}\phi_d(x)$. The factors are irreducible over the rationals.
In particular $$x^6-1=\phi_1(x)\phi_2(x)\phi_3(x)\phi_6(x)=(x-1)(x+1)(x^2+x+1)(x^2-x+1)$$
And $$\phi_6(x^n)=x^{2n}-... | {
"language": "en",
"url": "https://mathoverflow.net/questions/405105",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 1
} |
Growth of the "cube of square root" function Hello all, this question is a variant (and probably a more difficult one)
of a (promptly answered ) question that I asked here, at Is it true that all the "irrational power" functions are almost polynomial ?.
For $n\geq 1$, let $f(n)$ denote the "integer part"
(largest inte... | It's not too hard to put a bound on the size of second differences (since without the truncation, they are bounded above by a constant times $n^{-1/2}$), but getting the bound down to one seems delicate. It looks like it can be done with mindless brute force, though. I won't write all of the cases, but here is a star... | {
"language": "en",
"url": "https://mathoverflow.net/questions/18246",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 2,
"answer_id": 1
} |
Continuous or analytic functions with this property of sinc function This question is motivated by my previous post in SE (math.stackexchange.com).
Prove or disprove that $\frac{\sin x}{x}$ is the only nonzero entire (i.e. analytic everywhere), or continuous,
function, $f(x)$ on $\mathbb{R}$ such that $$\int_{-\infty}^... | Consider the function $f:\mathbb R\to\mathbb R$ which vanishes outside of $[-1,1]$ and such that for all $x \in [-1,1]$ has $$
f(x) = \frac{5 x^4}{4}+\frac{1}{2}
\sqrt{\frac{37}{6}} x^3-\frac{9
x^2}{4}-\frac{1}{2} \sqrt{\frac{37}{6}}
x+1.
$$
Then $$\int_\mathbb Rf(x)\,\mathrm dx=\int_\mathbb Rf(x)^2\,\mathrm d... | {
"language": "en",
"url": "https://mathoverflow.net/questions/48761",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 7,
"answer_id": 1
} |
the following inequality is true，but I can't prove it The inequality is
\begin{equation*}
\sum_{k=1}^{2d}\left(1-\frac{1}{2d+2-k}\right)\frac{d^k}{k!}>e^d\left(1-\frac{1}{d}\right)
\end{equation*}
for all integer $d\geq 1$. I use computer to verify it for $d\leq 50$, and find it is true, but I can't prove it. Thanks ... | This is true for large $d$, and probably for all $d$. I'll prove that the sum is
$$e^d(1-1/d+1/d^2+O(1/d^{2.5+\epsilon}))$$
and leave the explicit bounds to you.
Set $k=d+\ell$. For $|\ell| \leq d^{0.5+\epsilon}$, we have
$$1-1/(2d+2-k) = 1-\frac{1}{d} \frac{1}{1-(\ell-2)/d} = 1-\frac{1}{d} - \frac{\ell-2}{d^2} - \fr... | {
"language": "en",
"url": "https://mathoverflow.net/questions/133028",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "36",
"answer_count": 3,
"answer_id": 1
} |
Upper limit on the central binomial coefficient What is the tightest upper bound we can establish on the central binomial coefficients $ 2n \choose n$ ?
I just tried to proceed a bit, like this:
$ n! > n^{\frac{n}{2}} $
for all $ n>2 $. Thus,
$ \binom{2n}{n} = \frac{ (n+1) \ldots (2n) }{n!} < \frac{\left(\frac{\sum_{k=... | Here's a way to motivate and refine the argument that
Péter Komjáth attributes to Erdős.
Start by computing the ratio between the $n$-th and $(n-1)$-st
central binomial coefficients:
$$
{2n \choose n} \left/ {2(n-1) \choose n-1} \right.
= \frac{(2n)! \phantom. / \phantom. n!^2}{(2n-2)! \phantom. / \phantom. (n-1)^2}
= ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/133732",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 7,
"answer_id": 5
} |
A binomial determinant fomula Is there an existing or elementary proof of the determinant identity
$
\det_{1\le i,j\le n}\left( \binom{i}{2j}+ \binom{-i}{2j}\right)=1
$?
| I don't have a proof yet, but it's likely that this matrix has a simple LU decomposition which makes the determinant obviously $1$.
\begin{multline}
\scriptsize\begin{pmatrix} 1 & 1 & 1 & 1 & 1 & 1 \\\ 4 & 5 & 7 & 9 & 11 & 13 \\\ 9 & 15 & 28 & 45 & 66 & 91 \\\ 16 & 35 & 84 & 165 & 286 & 455 \\\ 25 & 75 & 210 & 495 & 1... | {
"language": "en",
"url": "https://mathoverflow.net/questions/154250",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 2,
"answer_id": 0
} |
Computing $\prod_p(\frac{p^2-1}{p^2+1})$ without the zeta function? We see that $$\frac{2}{5}=\frac{36}{90}=\frac{6^2}{90}=\frac{\zeta(4)}{\zeta(2)^2}=\prod_p\frac{(1-\frac{1}{p^2})^2}{(1-\frac{1}{p^4})}=\prod_p \left(\frac{(p^2-1)^2}{(p^2+1)(p^2-1)}\right)=\prod_p\left(\frac{p^2-1}{p^2+1}\right)$$
$$\implies \prod_p \... | I found a proof of
$$5 \sum_{m=1}^{\infty} \frac{1}{m^4} = 2 \left( \sum_{n=1}^{\infty} \frac{1}{n^2} \right)^2$$
by rearranging sums and wrote it up. The argument is just 1.5 pages, the other 4.5 are explanations and context.
Here is a summary using divergent sums; see the write up for a correct version. Set
$$h(m,n)... | {
"language": "en",
"url": "https://mathoverflow.net/questions/164092",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "42",
"answer_count": 4,
"answer_id": 1
} |
Is this system always solvable in radicals by quartics, octics, $12$-ics, etc? While considering this post, it made me wonder about its generalization in another direction and from the perspective of Galois theory.
Question: Is it true that, given four constants ($\alpha,\beta,\gamma,\delta$), then the system,
$$\beg... | For $(a,b,c,d)=(1,2,3,20)$ and $(\alpha,\beta,\gamma,\delta)=(1,1,1,0)$ one computes that $x_1$ is a root of the irreducible polynomial $4 x^{20} - 20 x^{19} + 40 x^{18} - 40 x^{17} + 195 x^{16} - 704 x^{15} + 1050 x^{14} - 700 x^{13} + 475 x^{12} - 900 x^{11} + 900 x^{10} - 300 x^{9} + 130 x^{8} - 260 x^{7} + 130 x^{6... | {
"language": "en",
"url": "https://mathoverflow.net/questions/227806",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
A question about (unicity of certain cycles in a Cayley graph of a) symmetric group Let $S=\{(1,2),(1,2,3,\ldots,n),(1,2,3,\ldots,n)^{-1}=(1,n\ldots,2)\}$ be a subset of the symmetric group $S_n$. We know that $(1,2,\ldots,n)(1,2)=(2,3,\ldots,n)$, and thus $$[(1,2,\ldots,n)(1,2)]^{n-1}=(1,2,\ldots,n)\overbrace{(1,2)\cd... | The smallest $n$ for which there exist sequences as asked for is $n = 7$:
*
*$(1,2,3,4,5,6,7) \cdot (1,2) \cdot (1,7,6,5,4,3,2) \cdot (1,2)
\cdot (1,2,3,4,5,6,7) \cdot (1,2) \cdot$
$(1,7,6,5,4,3,2) \cdot (1,2) \cdot (1,2,3,4,5,6,7) \cdot (1,2)
\cdot (1,7,6,5,4,3,2) \cdot (1,2) = ()$, and
*$(1,2,3,4,5,6,7... | {
"language": "en",
"url": "https://mathoverflow.net/questions/236791",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
seeking proofs: infinite series inequalities
Question. Numerically, the following is convincing. However, is there a proof?
$$\left(\sum_{k\geq1}\frac1{\sqrt{2^k+3^k}}\right)^4
<\pi^2\left(\sum_{k\geq1}\frac1{2^k+3^k}\right)\left(\sum_{k\geq1}\frac{k^2}{2^k+3^k}\right).$$
This comes up in some recent work and the i... | This may serve as a different approach. By Cauchy-Schwarz inequality, $$\left(\sum_{k\geq 1}\frac 1{\sqrt{2^k+3^k}}\right)^2\leq \left(\sum_{k\geq 1}\frac 1{k^2}\right)\left(\sum_{k\geq 1}\frac{k^2}{2^k+3^k}\right),$$ which shows that $$\left(\sum_{k\geq 1}\frac 1{\sqrt{2^k+3^k}}\right)^2\leq \frac{\pi^2}6\left(\sum_{k... | {
"language": "en",
"url": "https://mathoverflow.net/questions/262008",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
special values of symmetric functions at powers of $\frac1j$ Let $e_n(x_1,x_2,x_3,\dots)$ denote the $n$-th elementary symmetric function in the infinite variables $x_1,x_2,x_3,\dots$.
Let $u$ and $v$ be the roots of $z^2-6z+1=0$.
Question. Let $x_j=\frac1{j^8}$. The following seems to be true, but can one prove or di... | As Gro-Tsen suggests in the comments, we have to expand the infinite product $$f(t)=\prod_j \left(1-\frac{t^8}{j^8}\right)=\prod_{j;\,w^4=1} \left(1-\frac{\omega t^2}{j^2}\right)=\prod_{w^4=1}\frac{\sin\pi\sqrt{w}t}{\pi\sqrt{w}t},$$
we expand product of four sines as an alternating sum of cosines $$\sin a\sin b\sin c\s... | {
"language": "en",
"url": "https://mathoverflow.net/questions/265430",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
A problem of divisibility I came across the following problem. Find two integers, $u_{n}$ and $v_{n}$, such that
$$a_{n}=4u_{n}v_{n}+(6n-1)v_{n}+(6n-1)u_{n}+8n^{2}-4n$$
divides
$$b_{n}=-(2n-1)u_{n}v_{n}-(2n^{2}-3n)v_{n}-(2n^{2}-3n)u_{n}+8n^{2}.$$
For example, for $n=2$, the choices $u_{2}:=-2$ and $v_{2}:=0$ yield $... | If $a_n\mid b_n$, then $a_n$ also divides
$$Q := (2n-1)a_n + 4b_n = (2n+1)^2 (u_n + v_n+4n).$$
We will blatantly require $a_n = Q$. Notice that
$$a_n-Q = 4u_nv_n - 2(2n^2-n+1)u_n - 2(2n^2-n+1)u_n - 8n(2n^2+n+1) = (2u_n-(2n^2-n+1))(2v_n-(2n^2-n+1)) - (n+1)^2(2n+1)^2.$$
Thus, it's enough to take $d\mid (n+1)^2(2n+1)^2$ ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/290218",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Rational approximations of $\sqrt{2}$ in $\mathbb{R} \times \mathbb{Q}_7$ Note: this question was updated (2) after GNiklasch's answer was posted, and taking Gro-Tsen's comment into account. The initial question (1) dealt with $\mathbb{Q}_3$.
Original post (1). Let's try to solve the equation $x^2 - 2 = 0$ with $x = ... | Here is a full solution for the modified problem, inspired by Gro-Tsen's valuable comment.
1. There are infinitely many rational numbers $a/b\in\mathbb{Q}$ in lowest terms such that
$$ \left|\frac{a^2}{b^2}-2\right|_\infty\ll\frac{1}{b}\qquad\text{and}\qquad
\left|\frac{a^2}{b^2}-2\right|_7\ll\frac{1}{b}.$$
To see this... | {
"language": "en",
"url": "https://mathoverflow.net/questions/294684",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
Given a polynomial constraint equation in $n$ variables, can one conclude that the sum of the variables is non-negative? Currently I'm stuck as follows;
at least a positive proof if $n=3$ would be a great nice-to-have!
Consider real numbers $x_1,x_2,\dots,x_n$ satisfying
$$\prod^n_{k=1}\left(1-x_k^2\right)\:=\:\prod... | A counterexample: $n=4$,
$$(x_1,\dots,x_4)=\left(-\frac{1}{4},-\frac{3}{8},-\frac{3}{8},\frac{\sqrt{1970156929}-2048}{45375}\right).
$$
So, by what you noted, your conjecture fails to hold for any $n\ge4$.
The validity gap is indeed between $n=3$ and $n=4$.
Indeed, for $n=3$ Mathematica confirms your conjecture:
... | {
"language": "en",
"url": "https://mathoverflow.net/questions/342126",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Primality test for specific class of $N=8k \cdot 3^n-1$ This question is related to my previous question.
Can you prove or disprove the following claim:
Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$
Let $N=8k \cdot 3^n-1$ such that $n>2$ , $k>0$ , $8k <3^n$ and... | Assume that $N$ is prime. Then we prove $S_{n-2}\equiv 0\pmod N$, the assumption that $8k<3^n$ is not used. I do not know how to prove it in the opposite direction.
We have $P_m(2\cos t)=2\cos mt$, so they are Chebyshev polynomials and satisfy $P_{mn}=P_n\circ P_m$. Note that $x^3-3x=P_3$, thus $S_{i}=P_{18k\cdot 3^i}(... | {
"language": "en",
"url": "https://mathoverflow.net/questions/362151",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
A constant bizarrely related to the Fibonacci Numbers For roughly the past month, I have been studying denesting radicals. For example: the expression $\sqrt[3]{\sqrt[3]2-1}$ is a radical expression that contains another radical expression, so this radical is nested. Is there a way to express this with radicals that ar... | First of all we get $A=1-\sqrt[3]{\frac{1}{2}}+\sqrt[3]{\frac{1}{4}}=\frac{3\sqrt[3]{2}}{2(\sqrt[3]{2}+1)}$......(1)
And from here you can proof by induction...
$A^{F_n}=A^{F_{n-1}}.A^{F_{n-2}}$
If $A^{F_{n-1}}=(\frac{3}{2})^{F_{n-2}}(a_{n-1}+b_{n-1}\sqrt[3]{2}+c_{n-1}\sqrt[3]{4})$
and
$A^{F_{n-2}}=(\frac{3}{2})^{F_{n... | {
"language": "en",
"url": "https://mathoverflow.net/questions/362679",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
} |
Number of d-Calabi-Yau partitions This problem arises from algebraic geometry/representation theory, see https://arxiv.org/pdf/1409.0668.pdf (chapter 2).
We call a partition $p=[p_1,...,p_n]$ with $2 \leq p_1 \leq p_2 \leq ... \leq p_n$ d-Calabi-Yau (for $d \geq 1$) when $n-d-1=\sum\limits_{i=1}^{n}{\frac{1}{p_i}}$.
F... | Following up on @pbelmans's answer and researching an article mentioned in OEIS A007018, I believe your Question 2 was answered just under 100 years ago by David Curtiss (On Kellogg's Diophantine problem, Amer. Math. Monthly 29 (1922) 380-387).*
Curtiss confirms Kellogg's conjecture that the maximum $x_i$ in any $$\fra... | {
"language": "en",
"url": "https://mathoverflow.net/questions/379038",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 2,
"answer_id": 0
} |
Alternative proofs sought after for a certain identity Here is an identity for which I outlined two different arguments. I'm collecting further alternative proofs, so
QUESTION. can you provide another verification for the problem below?
Problem. Prove that
$$\sum_{k=1}^n\binom{n}k\frac1k=\sum_{k=1}^n\frac{2^k-1}k.$$
... | One can also use the binomial transform.
(If $A(z)=\sum_{i\geq 0} a_i z^i$ is a (formal) power series, the (formal) power series $B(z):=\frac{1}{1-z}
A(\frac{z}{1-z})$ has coefficients $[z^n] B(z)=\sum_{j=0}^n {n \choose j} a_j$).
We have $\log(\frac{1}{1-z})=\sum_{k\geq 1} \frac{z^k}{k}$.
Thus \begin{align*}
\sum_{k=1... | {
"language": "en",
"url": "https://mathoverflow.net/questions/379248",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 6,
"answer_id": 0
} |
What is the limit of $a (n + 1) / a (n)$? Let $a(n) = f(n,n)$ where $f(m,n) = 1$ if $m < 2 $ or $ n < 2$ and $f(m,n) = f(m-1,n-1) + f(m-1,n-2) + 2 f(m-2,n-1)$ otherwise.
What is the limit of $a(n + 1) / a (n)$? $(2.71...)$
| Here is a derivation for an explicit formula for $a(n)$.
The generating function for $f(m,n)$ is
$$F(x,y) := \sum_{m,n\geq 0} f(m,n)x^m y^n = \big(1 + \frac{3x^2y^2}{1-xy(1+2x+y)}\big)\frac{1}{1-x}\frac{1}{1-y}.$$
It follows that
\begin{split}
a(n) &= 1 + \sum_{i,j=0}^n [x^iy^j]\ \frac{3x^2y^2}{1-xy(1+2x+y)} \\
&= 1 + ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/389034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 2,
"answer_id": 0
} |
Definite integral of the square root of a polynomial ratio I found myself with the following integral
$$ \int_{b_1}^{b_2} \sqrt{\frac{(b-b_1)(b_2-b)(b_3-b)}{(b_4-b)}} \ db $$
with $ b_1 < b_2 < b_3 < b_4 $. I know that
$$ \int_{b_1}^{b_2} \frac{db}{\sqrt{(b-b_1)(b_2-b)(b_3-b)(b_4-b)}} $$
is equal to
$$ \frac{2}{(b_4-b_... | We may as well rescale and translate so $b_1 = 0$ and $b_2 = 1$, leaving just two parameters instead of four. Maple produces a result involving a limit of elliptic integrals:
$$
\frac{\underset{t \rightarrow 1-}{\mathrm{lim}}\frac{i \left(b_{3}^{2}+\left(2 b_{4}-2\right) b_{3}-3 b_{4}^{2}+2 b_{4}+1\right) \left(\sqrt{... | {
"language": "en",
"url": "https://mathoverflow.net/questions/422392",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Full-rank matrix I have a sparse square matrix and want to see if it is full rank (so that I can apply the implicit function theorem).
$$\left[\begin{array}{cccccccccc}
0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0\\
x_{1}^{2} & Nx_{1} & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\
0 & c & 0 & 0... | OK, let's call the block matrix above $M$. First eliminate $N$ by a substitution $c\mapsto d N$. Then substitute $z_i \mapsto d y_i$ to eliminate $d$. Then you can construct the Schur complement w.r.t. the first and last rows/columns of $M$ to get $\det(M) = -c^2 N x_1^2 x_2^2 x_3^2 \det P$, with
$$
P =
\begin{pmatrix... | {
"language": "en",
"url": "https://mathoverflow.net/questions/428308",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
An algebraic inequality in three real variables Is it true that
$$(v-u)^2+(w-u)^2+(w-v)^2 \\
+\left(\sqrt{\frac{1+u^2}{1+v^2}}
+\sqrt{\frac{1+v^2}{1+u^2}}\right) (w-u)(w-v) \\
-\left(\sqrt{\frac{1+u^2}{1+w^2}}+\sqrt{\frac{1+w^2}{1+u^2}}\right) (w-v)(v-u) \\
-\left(\sqrt{\frac{1+w^2}{1+v^2}}+\sqrt{\frac{1+v^2}{1+w^2}... | Rewrite the inequality in terms of $a,b,c>1$ via $u=(\frac{1}{a}-a)/2$, $v=(b-\frac{1}{b})/2$, and $w=(c-\frac{1}{c})/2$. Note that $\sqrt{1+u^2}=\frac{1+a^2}{2a}$ and so on. Then the conjectured inequality is equivalent to
\begin{equation}
((c - b)(a + c)(a + b))^2(a^2b^2c^2-a^2bc+ab^2c+abc^2+ab+ac-bc+1)>0.
\end{equa... | {
"language": "en",
"url": "https://mathoverflow.net/questions/435849",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Generalized Vieta-product It's known that
$$S_2={2\over\pi} = {\sqrt{2}\over 2}{\sqrt{2+\sqrt{2}}\over 2}{\sqrt{2+\sqrt{2+\sqrt{2}}}\over 2}\dots$$
The terms in the product approaches 1, the same holds for the following convergent series, with $\phi$ the golden ratio
$$S_1 = {\sqrt{1}\over\phi}{\sqrt{1+\sqrt{1}}\ov... | I doubt there exists a closed formula for $n\ne 2$. In the case $n=2$ such formula exists only thanks to the double-angle formula for cosine.
Let $n$ be fixed and $c=c_n$. Notice that $n=c^2-c$ and $c\to\infty$ as soon as $n\to\infty$.
Denote by $p_k$ the $k$-th multiplier in the product $S_n$. It can be easily seen th... | {
"language": "en",
"url": "https://mathoverflow.net/questions/62619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
} |
Exact Value of a Series It is very easy to show that the series
$$\frac{1-1/2}{1\times2} - \frac{1-1/2+1/3}{2\times3} + \frac{1-1/2+1/3-1/4}{3\times4} - ...$$
i.e.
$$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n(n+1)}[1-\frac{1}{2} + \frac{1}{3} - ...+ \frac {(-1)^{n}}{n+1}]$$
is convergent. Can one find its exact value? Or... | The interior sum is equal to $\int_0^1\frac{1-(-x)^{n+1}}{1+x}dx$ and
$$
\sum _{n=1}^{\infty } \frac{(-1)^{n+1} \left(1-(-x)^{n+1}\right)}{n (n+1) (x+1)}=
\frac{(x-1) \log (1-x)-x+\log (4)-1}{1+x}.
$$
So the answer is eqaul to
$$
\int_0^1 \frac{(x-1) \log (1-x)-x+\log (4)-1}{1+x}dx=\frac{\pi ^2}{6}+\log ^2(2)-2.
$$
| {
"language": "en",
"url": "https://mathoverflow.net/questions/75417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
Diophantine equation $2(x - 1/x) = y - 1/y$ Does the Diophantine equation $2(x - \frac{1}{x}) = y - \frac{1}{y}$ have only trivial rational solutions, i.e, $x=\pm1, y = \pm1$?
| This question is actually the first example, in a different formulation, of the following problem:
Find two integer right-angled triangles with a common base and altitudes in the integer ratio $N:1$,
which was considered by the late John Leech (in the 1980s I think).
We have to find integer solutions to
$B^2+A^2=C^2 \... | {
"language": "en",
"url": "https://mathoverflow.net/questions/76846",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 2
} |
Eigenvectors of asymmetric graphs Let $G$ be an asymmetric connected graph. Then is it always the case that at least one of the eigenvectors of its adjacency matrix $A$ consists entirely of distinct entries?
Thanks!
| I think the answer is no. Take the Frucht Graph, the simplest nontrivial asymmetric graph. Its adjacency matrix is
\begin{equation*}
\left(
\begin{array}{cccccccccccc}
0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 \\\\
1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\
0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \... | {
"language": "en",
"url": "https://mathoverflow.net/questions/104297",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Approximation of a given function by rational functions Given a function $1/\sqrt{x^2 -k^2}$ where k is a constant with a small imaginary part, how do you go about constructing a rational approximation? I am interested in the L_p (p=2 or $\infty$) norm of the difference being small on the real line. Both the theoretic... | Eleven years too late.
Using @Robert Israel's approach, the $[n,n]$ Padé approximants $P_n$
$$\sqrt s =\frac{1}{\sqrt{2}}\,\frac {1+\sum_{i=1}^n a_i\,\left(s-\frac{1}{2}\right)^i} {1+\sum_{i=1}^n b_i\,\left(s-\frac{1}{2}\right)^i}+O\left(\left(s-\frac{1}{2}\right)^{2n+1}\right) $$ can easily be built starting from the... | {
"language": "en",
"url": "https://mathoverflow.net/questions/144033",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Computing $\prod_p(\frac{p^2-1}{p^2+1})$ without the zeta function? We see that $$\frac{2}{5}=\frac{36}{90}=\frac{6^2}{90}=\frac{\zeta(4)}{\zeta(2)^2}=\prod_p\frac{(1-\frac{1}{p^2})^2}{(1-\frac{1}{p^4})}=\prod_p \left(\frac{(p^2-1)^2}{(p^2+1)(p^2-1)}\right)=\prod_p\left(\frac{p^2-1}{p^2+1}\right)$$
$$\implies \prod_p \... | A proof of this identity not using properties of the Riemann zeta function is listed as an unsolved problem in section B48 of Guy's Unsolved Problems in Number Theory.
An amusing observation: this identity implies the infinitude of primes. If the product over $p$ were finite, the final answer would be a rational number... | {
"language": "en",
"url": "https://mathoverflow.net/questions/164092",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "42",
"answer_count": 4,
"answer_id": 2
} |
For $x,y\ge 2$ does $x^4+x^2y^2+y^4$ ever divide $x^4y^4+x^2y^2+1$? For a problem in combinatorics, it comes down to knowing whether there exist integers $x,y\ge 2$ such that
$$
x^4+x^2y^2+y^4\mid x^4y^4+x^2y^2+1.
$$
Note that $x^6(x^2-y^2)(x^4+x^2y^2+y^4)+(x^2y^2-1)(x^4y^4+x^2y^2+1)=x^{12}-1$ and so we can look at a... | It seems to me that you need both $$p=x^2+xy+y^2$$ and $$q=x^2-xy+y^2$$ to divide $$r=x^4y^4+x^2y^2+1$$ Considering all expressions as polynomials in $x$, the remainder when you divide $r$ by $p$ is $$(y^7-y^3)x-y^4+1$$ and the remainder when you divide $r$ by $q$ is $$(y^3-y^7)x-y^4+1$$ if Maple and I are on the same... | {
"language": "en",
"url": "https://mathoverflow.net/questions/176351",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 1,
"answer_id": 0
} |
set of centers of sphere inscribed in tetrahedron Having a sphere and three diffrent point $A,B,C$ on this sphere. Find set of all centers of spheres inscribed in a tetrahedron $ABCD$, where $D$ is some point on the given sphere. The problem reduced to 2-dimensions is trivial it's just sum of two arcs of some circle, b... | Here is a depiction of Robert Bryant's surface defined by the $56$-term polynomial
he details:
Note one component is the "triangular tea bag" discernable in
my empirical investigation.
Here is the polynomial:
$$
2 x^5+x^4 y+x^4 z-7 x^4+2 x^3
y^2+4 x^3 y z-10 x^3 y+2
x^3 z^2-10 x^3 z+8 x^3+2
x^2 ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/214835",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
For which divisors $a$ and $b$ of $n$ does there exist a Latin square of order $n$ that can be partitioned into $a \times b$ subrectangles? There exists a Latin square of order $8$ which can be partitioned into $2 \times 4$ subrectangles:
$$
\begin{bmatrix}
\color{red} 1 & \color{red} 2 & \color{red} 3 & \color{red} 4 ... | I eventually co-authored a paper which includes this topic. The non-trivial results are:
*
*There exists a Latin square of order $n$ which decomposes into $2 \times (n/2)$ subrectangles for all even $n \not\in \{2,6\}$.
*There exists a Latin square of order n which decomposes into $3 \times (n/3)$ subrectangles if... | {
"language": "en",
"url": "https://mathoverflow.net/questions/229088",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
When $\frac{a^2}{b+c}＋\frac{b^2}{a+c}＋\frac{c^2}{a+b}$ is integer and $a,b,c$ are coprime natural numbers, is there a solution except (183,77,13)? Given $a,b,c\in \Bbb{N}$ such that $\{a,b,c\}$ are coprime natural numbers and $a,b,c>1$. When
$$\frac{a^2}{b+c}＋\frac{b^2}{c+a}＋\frac{c^2}{a+b}\in\mathbb Z\,?$$
I know the... | Yes, there is another solution. The next one I found is a bit big, namely
$$ a = 15349474555424019, b = 35633837601183731, c = 105699057106239769. $$
This solution also satisfies the property that
$$ \frac{a^{2}}{b+c} + \frac{b^{2}}{a+c} + \frac{c^{2}}{a+b} = \frac{31}{21} (a+b+c),
$$
which was true of $a = 13$, $b = 7... | {
"language": "en",
"url": "https://mathoverflow.net/questions/264335",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 1,
"answer_id": 0
} |
Proving an homogenous sextic product never a square Can anyone prove that $$2(a^2+b^2)(a^4+a^2b^2+b^4) \ne \square$$ for $a,b$ positive integers?
| I am a little confused. First, assume that $a, b$ are relatively prime (dividing through by the gcd divides the lhs by a square). Second, note that $a^2+b^2$ is relatively prime to $2(a^4 + b^2a^2+b^4)$ (since the expression within the last set of parentheses is $(a^2+b^2)^2 - a^2 b^2.$) So, if the thing has any hope o... | {
"language": "en",
"url": "https://mathoverflow.net/questions/265786",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Mean minimum distance for M and N uniformly random points on reals between 0 and 1 Similar to Mean minimum distance for N random points on a one-dimensional line, but instead of only N random points, choose N and M random points and find the mean minimum distance between points of N and M, but NOT M and M or N and N.
... | Let $D_1$ be the answer conditioned on the leftmost point being at 0 and the rightmost point at 1 (colors irrelevant). Let $0 = x_1 \leq \ldots \leq x_{n + m} = 1$ be the ordered coordinates of points. We can see that $s_i = x_{i + 1} - x_i$ are equidistributed subject to $s_1 + \ldots + s_{n + m - 1} = 1$, and colorin... | {
"language": "en",
"url": "https://mathoverflow.net/questions/285638",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Adventure with infinite series, a curiosity It is easily verifiable that
$$\sum_{k\geq0}\binom{2k}k\frac1{2^{3k}}=\sqrt{2}.$$
It is not that difficult to get
$$\sum_{k\geq0}\binom{4k}{2k}\frac1{2^{5k}}=\frac{\sqrt{2-\sqrt2}+\sqrt{2+\sqrt2}}2.$$
Question. Is there something similarly "nice" in computing
$$\sum_{k\geq... | Mathematica says, for the first:
$$\,
_4F_3\left(\frac{1}{8},\frac{3}{8},\frac{5}{8},\frac{7}{8};\frac{1}{4},\frac{1}{2},\frac{
3}{4};\frac{1}{4}\right)$$
and
$$\,
_8F_7\left(\frac{1}{16},\frac{3}{16},\frac{5}{16},\frac{7}{16},\frac{9}{16},\frac{11}{16}
,\frac{13}{16},\frac{15}{16};\frac{1}{8},\frac{1}{4},\... | {
"language": "en",
"url": "https://mathoverflow.net/questions/312307",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 4,
"answer_id": 3
} |
Irreducibility of a polynomial when the sum of its coefficients is prime I came up with the following proposition, but don't know how to prove it.
I used Maple to see that it holds when $ a + b + c + d <300 $.
Let $a,b,c$ and $d$ be non-negative integers such that $d\geq1$ and $a+b+c\geq1$.
If $a+b+c+d$ is a prime numb... | Francesco Polizzi's idea is enough to solve the problem:
Lemma. Let $f = ax^3 + bx^2 + cx + d \in \mathbf Z_{\geq 0}[x]$ nonconstant with $d > 0$, such that $f(1)$ is a prime $p > 2$. Then $f$ is irreducible in $\mathbf Z[x]$.
Proof. Suppose $f = gh$ for $g,h \in \mathbf Z[x]$; we must show that $g \in \{\pm 1\}$ or ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/365049",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Analytical solution to a specific differential equation I was wondering whether there is an analytical solution to the ODE
\begin{equation}
-n\int xy(x)dx + ihy'(x) + (x^2+k)y(x) = 0,
\end{equation}
where $n=0,1,2,...$, $h \in \mathbb{R}$, and $k=+1,0$ or $-1$.
For $n=0$ this can be solved exactly, and the solution is
... | For $k=0$ the solution is a hypergeometric function,
$$y(x)=C_1 \, _1F_1\left(\frac{2}{3}-\frac{n}{3};\frac{2}{3};\frac{i x^3}{3 h}\right)-(3h)^{-1/3}(-1)^{5/6} C_2 x \, _1F_1\left(1-\frac{n}{3};\frac{4}{3};\frac{i x^3}{3 h}\right),$$
which at least for some values of $n$ can be reduced to a Bessel function and/or an i... | {
"language": "en",
"url": "https://mathoverflow.net/questions/370503",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Do identities exist for the binomial series $\sum_{k=m+1}^{n+1} \binom{k}{m} \binom{n+1}{k-1} $? While examining the product of two upper triangular matrices, I've found that the $(m,n)$'th entry of the resulting matrix amounts to: $$\sum_{k=m+1}^{n+1} \binom{k}{m} \binom{n+1}{k-1} $$ when $n \geq m$ (all other entries... | $$\sum_{k}{{k\choose m} {n+1\choose k-1}}=\frac{1}{m}\sum_{k}k{k-1\choose m-1}{n+1\choose k-1} = \frac{n+1 \choose m-1}{m}\sum_{k}k{n-m+2 \choose k-m} = \frac{n+1 \choose m-1}{m}(\sum_{k}(k-m){n-m+2 \choose k-m}) + {n+1 \choose m-1}\sum_{k}{n-m+2 \choose k-m} = \frac{{n+1 \choose m-1}(n-m+2)}{m}(\sum_{k}{n-m+1 \choose ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/372066",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Throwing a fair die until most recent roll is smaller than previous one I roll a fair die with $n>1$ sides until the most recent roll is smaller than the previous one. Let $E_n$ be the expected number of rolls. Do we have $\lim_{n\to\infty} E_n < \infty$? If not, what about $\lim_{n\to\infty} E_n/n$ and $\lim_{n\to\inf... | The answer may be expressed more simply, in fact $E_n = \left( \frac{n}{n-1}\right)^n$.
Update 1: (The following was independently obtained by Pierre PC, I just found out after I finished typing.) The following is a simple way to see this. Let $E_{n,\ell}$ denote the expected number of rolls if the last roll is $\ell$.... | {
"language": "en",
"url": "https://mathoverflow.net/questions/376369",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 4,
"answer_id": 2
} |
Squares of the form $2^j\cdot 3^k+1$ Squares of the form $2^j\cdot 3^k+1$, for j,k nonnegative.
Is it known if there are infinitely many? And if $2^j\cdot 3^k+1=N$ is a square, then it must be necessarly a semi-prime?
Do you think that 3-smooth neighbour squares is a good name for these squares?
| We may find them all by elementary methods. Assume that $2^j3^k=(n-1)(n+1)$. Since $\gcd(n-1,n+1)\leqslant 2$, we get either $j=0$, $n-1=1$, $n+1=3$, or $j\geqslant 1$, $(n-1)/2$ and $(n+1)/2$ are consecutive 3-smooth numbers. They must be a power of 2 and a power of 3, so we should solve the equations $2^a=3^b\pm 1$. ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/383208",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
On Markoff-type diophantine equation Do there exist integers $x,y,z$ such that
$$
x^2+y^2-z^2 = xyz -2 \quad ?
$$
Why this is interesting? First, this equation arose in an answer to the previous Mathoverflow question What is the smallest unsolved diophantine equation? but was not asked explicitly as a separate question... | There is no solution.
Fix a solution $(x,y,z)$ with $|x|+|y|+|z|$ minimal. We will show a contradiction.
We can't have $xyz=0$ as we would then obtain one of the unsolvable equations $x^2+y^2= -2$, $x^2-z^2=-2$, $y^2-z^2=-2$.
If $xyz>0$, then by swapping the signs of two of $x,y,z$ if necessary we can assume $x,y,z>0$... | {
"language": "en",
"url": "https://mathoverflow.net/questions/392993",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
Monotonicity of a parametric integral For real $x>0$, let
$$f(x):=\frac1{\sqrt x}\,\int_0^\infty\frac{1-\exp\{-x\, (1-\cos t)\}}{t^2}\,dt.$$
How to prove that $f$ is increasing on $(0,\infty)$?
Here is the graph $\{(x,f(x))\colon0<x<3\}$:
So, it even appears that $f$ is concave. Since $f>0$, the concavity would of c... | Here is another solution: Noting the identity
$$\sum_{n=-\infty}^{\infty} \frac{1}{(t+2\pi n)^2} = \frac{1}{2(1-\cos t)}$$
and taking advantage of the fact that the integrand is non-negative, we may apply Tonelli's theorem to get
\begin{align*}
f(x)
&= \frac{1}{2\sqrt{x}} \int_{-\pi}^{\pi} \left( \sum_{n=-\infty}^{\inf... | {
"language": "en",
"url": "https://mathoverflow.net/questions/396552",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Does $\{\tau(1)\tau(2)+\cdots+\tau(n-1)\tau(n)+\tau(n)\tau(1):\ \tau\in S_n\}$ contain a unique multiple of $n^2$ for each $n\ge6$? Motivated by Question 397575, here I pose a related question.
Question.
Does the set $$T_n:=\{\tau(1)\tau(2)+\cdots+\tau(n-1)\tau(n)+\tau(n)\tau(1):\ \tau\in S_n\}$$ contain a unique multi... | No, for $n = 11$ this fails:
363 = 3 * 11^2 with [7, 2, 8, 5, 3, 4, 6, 9, 10, 1, 11]
484 = 2^2 * 11^2 with [10, 9, 6, 3, 1, 2, 4, 5, 7, 8,
11]
Running the code I wrote to check this a little more, there is more than one multiple of $n^2$ in the set you describe for all $11\leq n \leq 50$, see here for two permutati... | {
"language": "en",
"url": "https://mathoverflow.net/questions/397577",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How many cubes are the sum of three positive cubes? Are there infinitely many integer positive cubes $x^3 = a^3 + b^3 + c^3$ that are equal to the sum of three integer positive cubes? If not, how many of them are there?
| There are bivariate coprime polynomial parametrizations: https://sites.google.com/site/tpiezas/010
$$(a^4-2ab^3)^3 + (a^3 b+b^4)^3 + (2a^3 b-b^4)^3 = (a^4+a b^3)^3$$
Added If you drop the positivity constraint then there is another identity for $x=v^4$.
$$v^{12}=(v^4)^3=(9u^4)^3+(3uv^3-9u^4)^3+(v(v^3-9u^3))^3$$
I belie... | {
"language": "en",
"url": "https://mathoverflow.net/questions/413031",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
Splitting the integers from $1$ to $2n$ into two sets with products as close as possible For each positive integer $n$, split the integers $1$ to $2n$ into two sets of $n$ elements each, and such that the products of the elements in each of these sets are as close as possible, say they differ by $a(n)$.
It can be check... | Here are optimal solutions for $n \le 10$, and the two sets happen to be equicardinal even if you don't enforce that:
\begin{matrix}
n & a_n & \text{solution} \\
\hline
1 & 1 & \{2\},\{1\} \\
2 & 2 & \{2,3\},\{1,4\} \\
3 & 6 & \{1,5,6\},\{2,3,4\} \\
4 & 18 & \{1,5,6,7\},\{2,3,4,8\} \\
5 & 30 & \{2,3,4,8,10\},\{1,5,6,7,... | {
"language": "en",
"url": "https://mathoverflow.net/questions/419319",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
"answer_count": 1,
"answer_id": 0
} |
Writing $1-xyzw$ as a sum of squares Can you write $1 - xyzw$ in the form $p + q (1 - x^{2}-y^{2}-z^{2}-w^{2})$ where $p$ and $q$ are polynomials that are of the form $\sum g_{i}^{2}$ where $g_{i}$ $\in$ $\mathbb{R}[x,y,z,w]$?
For instance, in the two variable case, $1 - xy = \frac{1}{2} + \frac{1}{2}(x-y)^{2} + \frac{... | The question would be even more interesting if the factor $1-x^2-y^2-w^2-z^2$ was replaced by $4-x^2-y^2-w^2-z^2$. The reason is that a necessary condition for the existence of such $p,q$ is that whenever $xywz>1$, the last factor is also negative. And it turns out that
$$(xywz>1)\Longrightarrow(x^2+y^2+w^2+z^2<4),$$
... | {
"language": "en",
"url": "https://mathoverflow.net/questions/421332",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Full-rank matrix I have a sparse square matrix and want to see if it is full rank (so that I can apply the implicit function theorem).
$$\left[\begin{array}{cccccccccc}
0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0\\
x_{1}^{2} & Nx_{1} & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\
0 & c & 0 & 0... | Thanks very much to Fred Hucht for getting me to think about $c$ and its relationship with $N$.
The following is an approach for small $c$:
As suggested by the mention of the implicit function theorem, the variables $x_1,x_2,x_3,z_1,z_2,N$ are implicit solutions to a complicated set of equations and $c$ is a parameter.... | {
"language": "en",
"url": "https://mathoverflow.net/questions/428308",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Is it possible to simplify the coefficient matrix for large values of $x$? If I have a system of $8$ linear equations for the eight variables $\{\alpha ,\beta ,\gamma ,\delta ,\eta ,\lambda ,\xi ,\rho \}$ and with the three parameters $\{x,y,z\}$ reals and $x>0$. I want to work on nontrivial solutions of the system, th... | Let $M:=M(x,y,z)$ be the $8\times8$ matrix in question. Let $m(x):=M(x,0,0)$.
We have
$$\det m(x)=
-256 e^{i x/2} x^2 \cos (2 x) \big((x^2+1)^2 \cos (2 x)-(x^2-1)^2\big).$$
So, $|\det m(x)|$ will be oscillating between the value $0$ (attained when $\cos (2 x)=0$ -- that is, when $x=(2k-1)\pi/4$ for natural $k$) and the... | {
"language": "en",
"url": "https://mathoverflow.net/questions/433151",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Vanishing of a product of cyclotomic polynomials in characteristic 2 Let $n$ be a positive integer and $1\leq j\leq n$. Consider the following polynomial:
$$p_{n,j}(x)=\frac{\prod\limits_{i=1}^{n+1}\frac{x^{i}+1}{x+1}}{\prod\limits_{i=1}^{j}\frac{x^{i}+1}{x+1}\prod\limits_{i=1}^{n-j+1}\frac{x^{i}+1}{x+1}}\in\mathbb{F}[... | Claim: $p_{n,j}(1)$ vanishes modulo $2$ if and only there is a carry when adding $j$ and $n+1-j$ in base-$2$.
Proof: Let us write $x^n+1 \equiv x^n-1= \prod_{d \mid n} \Phi_d(x)$ where $\Phi_d$ is the $d$th cyclotomic polynomial. Then, since there are $\lfloor m/d\rfloor$ multiplies of $d$ in $\{1,2,\ldots,m\}$,
$$p_{n... | {
"language": "en",
"url": "https://mathoverflow.net/questions/438793",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Is there a good reason why $a^{2b} + b^{2a} \le 1$ when $a+b=1$? The following problem is not from me, yet I find it a big challenge to give a nice (in contrast to 'heavy computation') proof. The motivation for me to post it lies in its concise content.
If $a$ and $b$ are nonnegative real numbers such that $a+b=1$, sho... | Fixed now. I spent some time looking for some clever trick but the most unimaginative way turned out to be the best. So, as I said before, the straightforward Taylor series expansion does it in no time.
Assume that $a>b$. Put $t=a-b=1-2b$.
Step 1:
$$
\begin{aligned}
a^{2b}&=(1-b)^{1-t}=1-b(1-t)-t(1-t)\left[\frac{1}2b... | {
"language": "en",
"url": "https://mathoverflow.net/questions/17189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "29",
"answer_count": 8,
"answer_id": 0
} |
Exact Value of a Series It is very easy to show that the series
$$\frac{1-1/2}{1\times2} - \frac{1-1/2+1/3}{2\times3} + \frac{1-1/2+1/3-1/4}{3\times4} - ...$$
i.e.
$$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n(n+1)}[1-\frac{1}{2} + \frac{1}{3} - ...+ \frac {(-1)^{n}}{n+1}]$$
is convergent. Can one find its exact value? Or... | Since
$$\frac1{n(n+1)}=\frac1n-\frac1{n+1},$$
we have
$$\begin{aligned}
\sum_{n=1}^N\sum_{k=1}^{n+1}\frac{(-1)^{n+k}}{kn(n+1)}
&=\sum_{n=1}^N\sum_{k=1}^{n+1}\frac{(-1)^{n+k}}{kn}+\sum_{n=2}^{N+1}\sum_{k=1}^n\frac{(-1)^{n+k}}{kn}\\\\
&=2\sum_{n=1}^N\sum_{k=1}^n\frac{(-1)^{n+k}}{kn}-\sum_{n=1}^N\frac1{n(n+1)}-1+\sum_{k=1... | {
"language": "en",
"url": "https://mathoverflow.net/questions/75417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
A matrix diagonalization problem For matrices $X,Y\in [0,1]^{n\times m}$, for n > m, is there a square matrix $W\in R^{n\times n}$ so that $X^TWY$ is diagonal if and only if $Y = X$? Furthermore, $X$ and $Y$ are column normalized so that $X1_m = Y1_m = 1_m$, where $1_m$ is the m-length column vector with all entries eq... | UPDATE Sorry, previous version was wrong. For $n=m=2$, this computation shows that there is such a matrix. FURHTER UPDATE However, for $m=n=3$, it shows there isn't.
Taking $n=m=2$, we see that $X$ and $Y$ are of the form $\begin{pmatrix} x & 1-x \\ 1-x & x \end{pmatrix}$ and $\begin{pmatrix} y & 1-y \\ 1-y & y \end{p... | {
"language": "en",
"url": "https://mathoverflow.net/questions/92603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
When is the degree of this number 3? I am helping a friend of mine, that works in history of mathematics. She is studying the story of the solution of the cubic equation by Cardano. Sometimes she asks me some mathematical questions, that are very hard to motivate from a modern point of view, but that were interesting t... | Edit: The takeaway is that there is another, exclusive way to generate $a$ and $b$ that given an equation of degree $3$. First, chose parameters $a$ and $s$ such that $2a/s(s^2-3)$ is not a perfect cube and $1-4/s^2(s^2-3)^2$ is not a perfect square. This is most such values of $s$. Then choose $b$ according to the for... | {
"language": "en",
"url": "https://mathoverflow.net/questions/103967",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 0
} |
square root of a certain matrix Hello,
I'd like to know the square root of the following $n$ by $n$ matrix, for $n > 2$ and $r>0$:
$R_{ii}=r+1$ for $i < n$
$R_{ij}=r$ otherwise
The $2$ by $2$ case is given by
$\sqrt{R}=\frac{1}{d} \left[\begin{array}{cc} 1+r+\sqrt{r} & r \\\ r & r+\sqrt{r}\end{array}\right]$
where $d=\... | Let $P(a,b,c)$ be the $n\times n$ matrix where
$$
P(a,b,c)_{ij} = \begin{cases}
a & \text{ if } i,j < n \\
b & \text{ if } i < n \text{ and } j = n \\
b & \text{ if } i = n \text{ and } j < n \\
c & \text{ if } i = j = n.
\end{cases}
$$
If I understand correctly, you want the square root of $I+P(r,r,0)$. Yo... | {
"language": "en",
"url": "https://mathoverflow.net/questions/134013",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Computing $\prod_p(\frac{p^2-1}{p^2+1})$ without the zeta function? We see that $$\frac{2}{5}=\frac{36}{90}=\frac{6^2}{90}=\frac{\zeta(4)}{\zeta(2)^2}=\prod_p\frac{(1-\frac{1}{p^2})^2}{(1-\frac{1}{p^4})}=\prod_p \left(\frac{(p^2-1)^2}{(p^2+1)(p^2-1)}\right)=\prod_p\left(\frac{p^2-1}{p^2+1}\right)$$
$$\implies \prod_p \... | An elementary proof of the identity
$$ 2 \zeta(2)^2 = 5\zeta(4)$$
has been found by Don Zagier in the paper
http://people.mpim-bonn.mpg.de/zagier/files/tex/ConsequencesCohomologySL/fulltext.pdf
The idea seems similar to how David Speyer proves it in that he starts by defining the function
$$ f(m,n) = \frac{2}{n^3m} + ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/164092",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "42",
"answer_count": 4,
"answer_id": 3
} |
How does this sequence grow Let $a(n)$ be the number of solutions of the equation $a^2+b^2\equiv -1 \pmod {p_n}$, where $p_n$ is the n-th prime and $0\le a \le b \le \frac{p_n-1}2$. Is the sequence $a(1),a(2),a(3),\dots$ non-decreasing? Data for the first thousand values of the sequence supports this conjecture.
Here ... | The answer is yes, and the number of solutions with a prime $p$ is $\lfloor \frac{p+5}{8} \rfloor$ when $p \not\equiv 1 \pmod{8}$ and is $\lfloor \frac{p+5}{8} \rfloor + 1$ when $p \equiv 1 \pmod{8}$.
The equation $a^{2} + b^{2} + c^{2} = 0$ defines a conic in $\mathbb{P}^{2}/\mathbb{F}_{p}$. If $p > 2$ this conic has... | {
"language": "en",
"url": "https://mathoverflow.net/questions/171602",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Is $\lceil \frac{n}{\sqrt{3}} \rceil > \frac{n^2}{\sqrt{3n^2-5}}$ for all $n > 1$? An equivalent inequality for integers follows:
$$(3n^2-5)\left\lceil n/\sqrt{3} \right\rceil^2 > n^4.$$
This has been checked for n = 2 to 60000. Perhaps there is some connection to the convergents to $\sqrt{3}$.
$\lceil \frac{n}... | Equivalently, we want to know if
$\mathrm{sqceiling}(n^2/3) > n^4/(3n^2-5) = n^2/3 + 5/9 + 25/(27n^2) + ...$
where $\mathrm{sqceiling}()$ is the function taking a real to the next exact square.
This is equivalent to
$\mathrm{sqceiling}(n^2/3) - n^2/3 > 5/9 + 25/(27n^2) + ...$
but since $n^2/3$ is never a square for $n>... | {
"language": "en",
"url": "https://mathoverflow.net/questions/186419",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
This inequality why can't solve it by now (Only four variables inequality)? I asked a question at Math.SE last year and later offered a bounty for it, only johannesvalks give Part of the answer; A few months ago, I asked the author(Pham kim Hung) in Facebook, he said that now there is no proof by hand.and use of softw... | Here is a partial solution that reduces the problem to a (hopefully) simpler one.
The inequality is homogeneous, so we may assume that the RHS equals one.
Let
$$
S=\left\{x\in\mathbb R^4;\frac14\sum_ix_i^4=1,x_i>0\right\}
$$
and
$$
f(a,b,c,d)=\dfrac{1}{4}\left(\dfrac{a^2}{b}+\dfrac{b^2}{c}+\dfrac{c^2}{d}+\dfrac{d^2}{a}... | {
"language": "en",
"url": "https://mathoverflow.net/questions/193753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
"answer_count": 2,
"answer_id": 0
} |
Matrix Submodular Inequality Given $a,b,x > 0$ I know following the submodularity property holds:
\begin{align}
\frac{1}{a} - \frac{1}{a+x} \geq \frac{1}{a+b} - \frac{1}{a+b+x}
\end{align}
My question is, does this property hold for matrices? Precisely, for $A,B,X \succ 0$ is it the case that:
\begin{align}
A^{-1} - (A... | Consider
$$ A = \pmatrix{1 & 0\cr 0 & 1\cr},\ X = \pmatrix{1 & 0\cr 0 & 0\cr},\ B = \pmatrix{1 & 1\cr 1 & 1\cr}$$
$$ \eqalign{A^{-1} &- (A+X)^{-1} = \pmatrix{1/2 & 0\cr 0 & 0\cr}\cr &\not\succeq
(A+B)^{-1} - (A+X+B)^{-1} = \pmatrix{4/15 & -2/15 \cr -2/15 & 1/15}}$$
Yes, I know $B$ and $X$ are positive semidefinite ra... | {
"language": "en",
"url": "https://mathoverflow.net/questions/196094",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$2^n$-1 consisting only of small factors I've checked the factorization of $2^N - 1$ up through N = 120 for the largest prime factor, and it looks like the largest value of N where $2^N-1$ has a largest prime factor under 2500 is N = 60 (largest prime factor = 1321). As N gets larger, the largest prime factors get larg... | It is true that if $N > 60$, then $2^{N} - 1$ has a prime factor $> 2500$.
Here's another approach. First, observe that every prime factor of $2^{p} - 1$ is $\equiv 1 \pmod{p}$. Combining this with the observation that if $a | b$, then $2^{a} - 1 | 2^{b} - 1$, we see that if $2^{N} - 1$ has all prime factors $\leq 2500... | {
"language": "en",
"url": "https://mathoverflow.net/questions/199599",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Product of a Finite Number of Matrices Related to Roots of Unity Does anyone have an idea how to prove the following identity?
$$
\mathop{\mathrm{Tr}}\left(\prod_{j=0}^{n-1}\begin{pmatrix}
x^{-2j} & -x^{2j+1} \\
1 & 0
\end{pmatrix}\right)=
\begin{cases}
2 & \text{if } n=0\pmod{6}\\
1 & \text{if } n=1,5\pmod{6}\\
-1 ... | UPDATE: As pointed out by Peter Mueller in the comments below, this reduction to continuants does not quite work as stated. I still believe that there is some connection with (possibly generalized) continuants and leave this answer as an unsuccessful attempt to demonstrate it (in hope to fix it later).
Notice that
$$\b... | {
"language": "en",
"url": "https://mathoverflow.net/questions/213246",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 1
} |
asymptotic for the number of involutions in GL(n,2) Is it known how the number of involutions in $GL_n(2)$, the group of $n\times n$ matrices over $\mathbb{Z}/2\mathbb{Z}$, behaves as $n\to\infty$ ?
Equivalently, one may ask this for the number of $n\times n$ matrices $A$ over $\mathbb{Z}/2\mathbb{Z}$ satisfying $A^2=0... | We may write $|{\rm GL}(n,2)| = 2^{n^{2}} \prod_{j=1}^{n}( 1- \frac{1}{2^{j}}).$
As $n \to \infty$, the rightmost factor tends to $\left( \sum_{r=0
}^{\infty} \frac{p(r)}{2^{r}} \right)^{-1}$, where $p(r)$ is the number of partitions of $r$. Let us write $|{\rm GL}(n,2)| = 2^{n^{2}}f(n)$.
Then we see that the number of... | {
"language": "en",
"url": "https://mathoverflow.net/questions/219615",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
} |
Product of arithmetic progressions Let $(a_1,a_2\ldots,a_n)$ and $(b_1,b_2,\ldots,b_n)$ be two permutations of arithmetic progressions of natural numbers. For which $n$ is it possible that $(a_1b_1,a_2b_2,\dots,a_nb_n)$ is an arithmetic progression?
The sequence is (trivially) an arithmetic progression when $n=1$ or $2... | First, it's easy to see that real solutions can be rescaled to be rational and therefore also integer.
For $n=7$ there are no solutions.
For $n=6$ there are essentially 4 primitive integral solutions, with only one consisting of natural numbers (as requested):
$\big(-35+12(1,2,4,3,6,5)\big) \times \big(-5+2(5,6,1,2,3,4... | {
"language": "en",
"url": "https://mathoverflow.net/questions/312896",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
Does the equation $(xy+1)(xy+x+2)=n^2$ have a positive integer solution? Does there exist a positive integral solution $(x, y, n)$ to $(xy+1)(xy+x+2)=n^2$? If there doesn't, how does one prove that?
| It looks that Vieta jumping helps.
For fixed positive integer $y$ choose a minimal positive integer $x$ for which $(xy+1)(xy+x+2)$ is a perfect square.
Denote $4(xy+1)(xy+x+2)=4n^2=(2xy+x+3-z)^2$ for some integer $z=2n-2xy-x-3$, this yields $0<z<x+3$ and rewrites as $z^2-2z(2xy+x+3)+x^2+2x+1=0$. Note that $x$ must div... | {
"language": "en",
"url": "https://mathoverflow.net/questions/313339",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 3,
"answer_id": 0
} |
Number of integer partitions modulo 3 Motivated by Parity of number of partitions of $n!/6$ and $n!/2$, I asked my computer (and my FriCAS package for guessing) for an algebraic differential equation for the number of integer partitions modulo 3. This is what it answered:
(1) -> s := [partition n for n in 1..]
(1... | Well, $f''f^2 +xf' ^3+2ff'^2 =0$ modulo 3 for $f=\prod(1-x^m)=\sum_{n\in \mathbb{Z} } (-1)^nx^{n(3n+1)/2}$ may be quickly seen as follows.
Differentiating the power series for $f$ and expanding the brackets we see that we should prove that
$$
\sum_{a(3a+1)/2+b(3b+1)/2+c(3c+1)/2=n}
(-1)^{a+b+c} \left(a(a-2)-abc+2ab\rig... | {
"language": "en",
"url": "https://mathoverflow.net/questions/323343",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
How can one integrate over the unit cube, subject to certain (quantum-information-theoretic) constraints? To begin, we have two constraints
\begin{equation}
C1=x>0\land z>0\land y>0\land x+2 y+3 z<1
\end{equation}
and
\begin{equation}
C2=x>0\land y>0\land x+2 y+3 z<1\land x^2+x (3 z-2 y)+(y+3 z)^2<3 z.
\end{equation}
$... | Well, maybe the second comment of LSpice made me fully realize that the constraint $C4$ means that there is only one degree of freedom between $a,b,c$. So, as a start I took $a=\frac{1}{3}$. Then, using the Mathematica GenericCylindricalDecomposition command and choosing the order of integration of $x,y,z$ to select th... | {
"language": "en",
"url": "https://mathoverflow.net/questions/331817",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
how to prove an equation involving sums of Kronecker symbol Let $p\equiv 8 \mod 9$ be a prime, I find the following equation:
$$2\sum_{\substack{0<x<p\\ 2|x}}\sum_{r|p^2-x^2}\left(\frac{-3}{r}\right)=p+1.$$
where $\left(\frac{-3}{r}\right)$ is the Kronecker symbol.
I checked it for many $p$ using computer. Does anyone ... | The identity can be rewritten as
$$\sum_{\substack{|x|<p\\ 2|x}}\sum_{r|p^2-x^2}\left(\frac{-3}{r}\right)=p+2,$$
because for $x=0$ the inner sum is $1-1+1=1$. Writing $x=2c$, the identity becomes
$$\sum_{|c|<p/2}\,\sum_{r|p^2-4c^2}\left(\frac{-3}{r}\right)=p+2.$$
The inner sum counts the number of integral representati... | {
"language": "en",
"url": "https://mathoverflow.net/questions/346596",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
How to determine the coefficient of the main term of $S_{k}(x)$? Let $k\geqslant 2$ be an integer, suppose that $p_1,p_2,\dotsc,p_k$ are primes not exceeding $x$. Write
$$ S_{k}(x) = \sum_{p_1 \leqslant x} \dotsb \sum_{p_k \leqslant x} \frac{1}{p_1+\dotsb +p_k}. $$
By AM-GM inequality, $p_{1}+\dotsb + p_{k} \geqslant ... | Thank you, Mr. Petrov, but you made a little mistake.
A detailed calculation of $c$ is as follows:
Write $g(x)=(1-\mathrm{e}^{-x})^k= \sum\limits_{j=0}^{k} \binom{k}{j} (-1)^{j} \mathrm{e}^{-jx}$, integrating by parts we get
\begin{align}
\int_{0}^{\infty} g(x) x^{-k} \,\mathrm{d} x & = \int_{0}^{\infty} g(x) \,\mathrm... | {
"language": "en",
"url": "https://mathoverflow.net/questions/364246",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Integrality of a binomial sum The following sequence appears to be always an integer, experimentally.
QUESTION. Let $n\in\mathbb{Z}^{+}$. Are these indeed integers?
$$\sum_{k=1}^n\frac{(4k - 1)4^{2k - 1}\binom{2n}n^2}{k^2\binom{2k}k^2}.$$
POSTSCRIPT. After Carlo's cute response and several useful comments, I like to ... | There is a way to prove Zhi-Wei Sun's identity as well as Carlo Beenakker's identity. Of course, both can be treated in accord with Fedor Petrov's induction. Let's focus on Sun's identity. Divide through by $\binom{2n}n\binom{3n}n$ to write
$$A_n:=\sum_{k=1}^n\frac{(9k-2)27^{k-1}}{k^2\binom{2k}k\binom{3k}k}=\frac{27^n}... | {
"language": "en",
"url": "https://mathoverflow.net/questions/374414",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 1
} |
Alternative proofs sought after for a certain identity Here is an identity for which I outlined two different arguments. I'm collecting further alternative proofs, so
QUESTION. can you provide another verification for the problem below?
Problem. Prove that
$$\sum_{k=1}^n\binom{n}k\frac1k=\sum_{k=1}^n\frac{2^k-1}k.$$
... | Here's a sketch of a proof of a generalization:
\begin{multline}
\quad
\sum_{k=1}^n\binom nk \frac{t^k}{k+a}\\ =\frac{1}{\binom{a+n}{n}}\sum_{k=1}^n \binom {a+k-1}{k-1} \frac{(1+t)^k-1}{k}. \quad
\tag {$*$}
\end{multline}
(This is a generalization of Terry Tao's generalization, which is the case $a=0$.)
We start with t... | {
"language": "en",
"url": "https://mathoverflow.net/questions/379248",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 6,
"answer_id": 2
} |
Is the solution to this trig function known to be algebraic or transcendental? This largest solution to this gorgeous equation is the first local extremum on a function related to the Fibonacci sequence:
$$x^2 \cdot \sin \left(\frac{2\pi}{x+1} \right) \cdot \left(3+2 \cos \left(\frac{2\pi}{x} \right) \right) = (x+1)^2 ... | It should be possible to show that $x$ is irrational using Theorem 7 of Trigonometric diophantine equations (On vanishing sums of roots of unity) by J. H. Conway and A. J. Jones, Acta Arithmetica 30 (1976), 229–240, although I have not carried out the full calculation. By letting $\alpha = 2\pi/(x+1)$ and $\beta = 2\p... | {
"language": "en",
"url": "https://mathoverflow.net/questions/385225",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
How can I find all integer solutions of $3^n - x^2 = 11$ I know that $n$ can't be even because of the following argument:
Let $n = 2p$. Then we can use the difference of two squares and it becomes like this :
$(3^p + x)(3^p - x) = 11; 3^p + x = 11 , 3^p - x = 1$. $3^p = 6$ which is not possible if $p$ is an integer.
I ... | $$3^{n} - x^2 = 11$$
According to Silverman's answer, we take the three cases $n=3a, n=3a+1,$ and $n=3a+2.$
The problem can be reduced to finding the integer points on elliptic curves as follows.
$\bullet n=3a$
Let $X = 3^a, Y=x$, then we get $Y^2 = X^3 - 11.$
According to LMFDB, this elliptic curve has integer points... | {
"language": "en",
"url": "https://mathoverflow.net/questions/401848",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Integers representable as binary quadratic forms It is known that odd prime $p$ can be represented as $p=x^2+y^2$ if and only if $p \equiv 1$ mod $4$, represented as $p=x^2+2y^2$ if and only if $p \equiv 1$ or $3$ mod $8$, represented as $p=x^2+3y^2$ if and only if $p \equiv 0$ or $1$ mod $3$, etc. The situation with r... | you don't seem to be mentioning Gauss composition. You have a genus of forms, equivalent to $\langle 1,8,27 \rangle,$ then $\langle 3,8,29 \rangle,$ then $\langle 9,8,3 \rangle.$ These are convenient for Dirichlet's description of composition.
There is a cancellation for principal forms: if $x^2 + 8xy + 27 y^2$ repr... | {
"language": "en",
"url": "https://mathoverflow.net/questions/418853",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What are the integer solutions to $z^2-y^2z+x^3=0$? The question is to describe ALL integer solutions to the equation in the title. Of course, polynomial parametrization of all solutions would be ideal, but answers in many other formats are possible. For example, answer to famous Markoff equation $x^2+y^2+z^2=3xyz$ is ... | Since the equation is NOT homogeneous, it is trivial to find
infinite families of solutions with $g=\gcd(x,y)>1$. For instance,
choose any integers $a$ and $b$, set $c=ab^2-a^2$, so multiplying
by $c^8$ gives $(ac^4)^2-ac^4(bc^2)^2+c^9=0$, giving the trivial
solutions $(x,y,z)=(c^3,bc^2,ac^4)$, but there are many other... | {
"language": "en",
"url": "https://mathoverflow.net/questions/430242",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 1
} |
$3^n - 2^m = \pm 41$ is not possible. How to prove it? $3^n - 2^m = \pm 41$ is not possible for integers $n$ and $m$. How to prove it?
| As a valuable hint for solving the problem, I consider the following extract from my lectures on elementary number theory.
Theorem ($\approx1320$; Levi ben Gerson 1288--1344).
The equations
$$
(1) \quad 3^p-2^q=1
$$
and
$$
(2) \quad 2^p-3^q=1
$$
have no solutions in integers $p,q>1$, except the solution
$p=2$, $q=3$ to... | {
"language": "en",
"url": "https://mathoverflow.net/questions/29926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 1
} |
Sharp upper bounds for sums of the form $\sum_{p \mid k} \frac{1}{p+1}$ Are there known sharp upper bounds (in terms of $k$ or $\omega(k)$, the number of distinct prime divisors of $k$) for sums of the form $\sum_{p \mid k} \frac{1}{p+1}$ for $k > 1$ subject to the constraint $\sum_{p \mid k} \frac{1}{p+1} < 1$? (This... | If we are allowed to consider somewhat weaker bounds, both answers depend only on the number of unitary divisors of $k$, which is $2^{\omega(k)}$. By the quoted result of O. Izhboldin and L. Kurliandchik (see Fedor's and Myerson's comments), for any set of $n$ positive integers {$a_{1}, \dots, a_{n}$} such that $\sum_{... | {
"language": "en",
"url": "https://mathoverflow.net/questions/43326",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to decide this function takes integer values? I am working a combinatorial question. I need to decide when the following function
$$\frac{m(7m^2 - 22m +7)}{ \sqrt{ (m^2- 2m + 9)(9m^2 - 2m +1) } } $$takes integer values with following
assumption: 1) m takes all positive integer values; 2) $(m^2- 2m + 9)(9m^2 - 2m... | An elementary approach is enough here. If the given ratio is an integer, then any prime $p$ which divides $m^2 -2m+9$ divides either $m$ or $7m^2-22m+7$. If $p$ divides $m$ then clearly $p =3$. If $p$ divides $7m^2-22m+7$ then $p$ divides $$[7m^2-14m+63 - (7m^2-22m+7)] = 8m+56. $$
Hence either $p=2$ or $p$ divides $m+7... | {
"language": "en",
"url": "https://mathoverflow.net/questions/66873",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 3,
"answer_id": 1
} |
Building a genus-$n$ torus from cubes I wonder if this has been studied:
What is the fewest number of unit cubes
from which one can build an $n$-toroid?
The cubes must be glued face-to-face,
and the boundary of the resulting object should
be topologically equivalent to an $n$-torus, by which I mean
a genus-$n$ han... | Theorem. Let $c(g)$ be the minimum number of cubes such that the boundary of some configuration of $c(g)$ cubes is a genus $g$ surface. Then $c(g)/g \to 2$ as $g \to \infty$.
Proof. We write $\chi(X)$ for the compactly supported Euler characteristic
of $X$, i.e.,
$\chi(X) = \sum (-1)^i \dim H^i_c(X, \mathbb{Q})$.
... | {
"language": "en",
"url": "https://mathoverflow.net/questions/98925",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
"answer_count": 1,
"answer_id": 0
} |
When is the degree of this number 3? I am helping a friend of mine, that works in history of mathematics. She is studying the story of the solution of the cubic equation by Cardano. Sometimes she asks me some mathematical questions, that are very hard to motivate from a modern point of view, but that were interesting t... | $$t:= \sqrt[3]{d(1 + x^6) + \sqrt{d^2 (1 - x^6)^2}} + \sqrt[3]{d(1 + x^6) - \sqrt{d^2 (1 - x^6)^2}}
+ \sqrt{d^2(1 +x^6)^2 - d^2(1 - x^6)^2}$$
$$= \sqrt[3]{2 d} + \sqrt[3]{2 d x^6} + \sqrt{4 d^2 x^6}$$
$$
= (1 + x^3) \sqrt[3]{2d} + 2 d x^3;$$
$$\frac{p^3}{27} = \left(\frac{q^2}{4} + \frac{p^3}{27}\right) - \left(\fra... | {
"language": "en",
"url": "https://mathoverflow.net/questions/103967",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 1
} |
Is $\lceil \frac{n}{\sqrt{3}} \rceil > \frac{n^2}{\sqrt{3n^2-5}}$ for all $n > 1$? An equivalent inequality for integers follows:
$$(3n^2-5)\left\lceil n/\sqrt{3} \right\rceil^2 > n^4.$$
This has been checked for n = 2 to 60000. Perhaps there is some connection to the convergents to $\sqrt{3}$.
$\lceil \frac{n}... | A tight result is $$\lceil \frac{n}{\sqrt{3}}\rceil \ge \frac{n^2}{\sqrt{3n^2-6+\frac{12}{n^2+2}}}$$
This means that the desired inequality holds for $n \ge 4.$ The cases $n=2,3$ can be checked separately.
As noted elsewhere, for $m=\lceil \frac{n}{\sqrt{3}}\rceil$ we have $n^2 \le 3m^2-2$ with equality exactly for t... | {
"language": "en",
"url": "https://mathoverflow.net/questions/186419",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
This inequality why can't solve it by now (Only four variables inequality)? I asked a question at Math.SE last year and later offered a bounty for it, only johannesvalks give Part of the answer; A few months ago, I asked the author(Pham kim Hung) in Facebook, he said that now there is no proof by hand.and use of softw... | As to why the question is hard - one could reformulate it as $$\frac{x+y+z+t}{4}\stackrel{?}{\ge} \sqrt[4]{\frac{(x^8 y^4 z^2 t)^{4/15}+(y^8 z^4 t^2 x)^{4/15}+(z^8 t^4 x^2 y)^{4/15}+(t^8 x^4 y^2 z)^{4/15}}{4}}$$
where $x=a^2/b$, $y=b^2/c$, $z=c^2/d$ and $t=d^2/a$. Then there is a tug of war between
$$\frac{x+y+z+t}{4}\... | {
"language": "en",
"url": "https://mathoverflow.net/questions/193753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
"answer_count": 2,
"answer_id": 1
} |
Infinitely many $k$ such that $[a_k,a_{k+1}]>ck^2$ Let $a_n\in \mathbf{N}$ be an infinite sequence such that $\forall i\neq j, a_i\neq a_j$.
I have the following theorem:
For $0<c<\frac{3}{2}$, there are infinitely many $k$ for which $[a_k,a_{k+1}]>ck$, where $[\cdots]$ denotes least common multiple.
Idea of proof: B... | In fact there are sequences $\{a_k\}$ of pairwise distinct positive integers
such that $[a_k, a_{k+1}] \ll k^{1+\epsilon}$ for all $\epsilon > 0$.
We first exhibit a sequence with
$[a_k, a_{k+1}] \ll k^{3/2} \log^3 k$ for all $k>1$, which already disproves
the conjecture that $[a_k, a_{k+1}] \gg k^2$ infinitely often.
... | {
"language": "en",
"url": "https://mathoverflow.net/questions/228762",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 3,
"answer_id": 1
} |
odd length Chevalley relations (in rank two) The unipotent radicals $\text{N}$ of the Borel subgroups of the complex algebraic groups of type $A_2$, $B_2$, and $G_2$ can each be abstractly presented using two one-parameter subgroups $x_1, x_2: \Bbb{C} \longrightarrow \text{N}$ subject to
the following defining relatio... | Lieber Señor Leverkühn,
Instead of using the Auslander ring
$\Bbb{C}\big[ t \big] \Big/ \Big( t^d = 0 \Big)$ use a two
variable version $\mathcal{L}_{2+3} = \Bbb{C}\big[s,t \big]\Big/ \Big(s^3t^2 = s^2t^3 = 0 \Big)$ and form the truncated "double" loop group $\text{SL}_2 \Big( \mathcal{L}_{2+3} \Big)$. As in your cons... | {
"language": "en",
"url": "https://mathoverflow.net/questions/229904",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Prove this conjecture inequality This following problem is from my Conjecture many years ago,
Question :
Let $a,b>0,n\in N^{+},n\ge 3$,such
$$a^n+b^n+(2n+2)(ab)^n\le 2n$$
Conjecture: then $a+b\le 2$
or
$a+b>2.a>0.b>0,n\ge 3$,then we prove
$$a^n+b^n+(2n+2)a^nb^n-2n>0$$
or
it suffuce to prove
$$a^n+b^n+(2n+2)a^nb^n-2... | Take $a=1+x,b=1-x$, $|x|\leqslant 1$. We have to prove $(1+x)^n+(1-x)^n+(2n+2)(1-x^2)^n\geqslant 2n$. By Bernoulli inequality we have $(1-x^2)^n\geqslant 1-nx^2$ and by binomial expansion $(1+x)^n+(1-x)^n=2+n(n-1)x^2+2\binom{n}4 x^4+\dots$
So, it suffices to prove that $$2+n(n-1)x^2+2\binom{n}4 x^4+2n+2-n(2n+2)x^2\geqs... | {
"language": "en",
"url": "https://mathoverflow.net/questions/238628",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Another question about the golden ratio and other numbers This is an extension of "A question about the golden ratio and other numbers." Given $r$, suppose that $$c_0+c_1x+c_2x^2+ \cdots = \frac{1} {\lfloor{r}\rfloor+\lfloor{2r}\rfloor x+\lfloor{3r}\rfloor x^2+ \cdots}.$$
Let $L(r) = \lim_{i\to\infty} \frac{c_{i+1}}{... | Some observations, but not a solution yet.
Let $t_k=\frac{F_{k+1}}{F_k}$ where $F_k$ is the $k^{th}$-Fibonacci number. The convention here for $F_k$ is that $F_3=2, F_4=3, F_5=5, \dots$. Denote the RHS in the above series by
$$\Psi_k(x)=\left(\sum_{n=1}^{\infty}\lfloor nt_k\rfloor x^{n-1}\right)^{-1}.$$
Then, it seems ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/259821",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
The average of reciprocal binomials This question is motivated by the MO problem here. Perhaps it is not that difficult.
Question. Here is an cute formula.
$$\frac1n\sum_{k=0}^{n-1}\frac1{\binom{n-1}k}=\sum_{k=1}^n\frac1{k2^{n-k}}.$$
I've one justification along the lines of Wilf-Zeilberger (see below). Can you ... | Let me elaborate on Fry's suggestion and your forthcoming comment.
$$\frac1{2^{n+1}}\sum_{k=1}^{n+1}\frac{2^k}k=\frac1{2^{n+1}}\int_0^2(1+x+\dots+x^n)dx=\frac1{2^{n+1}}\int_0^2\frac{1-x^{n+1}}{1-x}dx=\\2\int_{0}^1\frac{(1/2)^{n+1}-t^{n+1}}{1-2t}dt=2\int_{0}^1\frac{(1-s)^{n+1}-(1/2)^{n+1}}{1-2s}ds.$$
We used change of v... | {
"language": "en",
"url": "https://mathoverflow.net/questions/262578",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
Subsets and Splits
Fractions in Questions and Answers
The query retrieves a sample of questions and answers containing the LaTeX fraction symbol, which provides basic filtering of mathematical content but limited analytical insight.