Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
$\pi$ in terms of polygamma The computer found this, but couldn't prove it.
Let $\psi(n,x)$ denote the polygamma function.
With precision 500 decimal digits we have:
$$ \pi^2 = \frac{1}{4}(15 \psi(1, \frac13) - 3 \psi(1, \frac16)) $$
Is it true?
In machine readable form:
pi^2 == 1/4*(15*psi(1, 1/3) - 3*psi(1, 1/6))
| Note that
$$
\psi(m,x) =(-1)^{m+1} m! \sum_{k=0}^{\infty} \frac{1}{(x+k)^{m+1}}.
$$
Therefore
$$
\psi(m,1/6) = (-1)^{m+1} m! \sum_{k=0}^{\infty} \frac{1}{(k+1/6)^{m+1}} =(-1)^{m+1} m! 6^{m+1} \sum_{n\equiv 1 \mod 6} \frac{1}{n^{m+1}}.
$$
Writing the condition $n\equiv 1 \mod 6$ as $n\equiv 1 \mod 3$ but not $4 \... | {
"language": "en",
"url": "https://mathoverflow.net/questions/312479",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Permutations $\pi\in S_n$ with $\sum_{k=1}^n\frac1{k+\pi(k)}=1$ Let $S_n$ be the symmetric group of all the permutations of $\{1,\ldots,n\}$.
Motivated by Question 315568 (http://mathoverflow.net/questions/315568), here I pose the following question.
QUESTION: Is it true that for each integer $n>5$ we have $$\sum_{k=1... | Claim: $a_n>0$ for all $n\geq 6\quad (*)$.
Proof: We use induction to prove $(*)$.
We have $a_6,a_7,a_8,a_9,a_{10},a_{11}>0$. Assume $(*)$ holds for all the integers $\in [6,n-1]$. We want to show that $a_n>0$ for all $n\geq12$.
If $n$ is an odd, let $n=2m+1$, we have $m\geq6$, so by induction hypothesis, there exist... | {
"language": "en",
"url": "https://mathoverflow.net/questions/315648",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 1,
"answer_id": 0
} |
Maximum eigenvalue of a covariance matrix of Brownian motion $$ A := \begin{pmatrix}
1 & \frac{1}{2} & \frac{1}{3} & \cdots & \frac{1}{n}\\
\frac{1}{2} & \frac{1}{2} & \frac{1}{3} & \cdots & \frac{1}{n}\\
\frac{1}{3} & \frac{1}{3} & \frac{1}{3} & \cdots & \frac{1}{n}\\
\vdots & \vdots & \vdots & \ddots & \frac{1}{n}\\
... | In this answer I show that the largest eigenvalue is bounded by $5< 3 + 2\sqrt{2}$. I will first use the interpretation of this matrix as the covariance matrix of the Brownian motion at times $(\frac{1}{n},\dots, 1)$ (I reversed the order so that the sequence of times is increasing, which is more natural for me).
We ha... | {
"language": "en",
"url": "https://mathoverflow.net/questions/366339",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Is this Laurent phenomenon explained by invariance/periodicity? In Chapter 4 (page 23, subsection "Somos sequence update") of his Tracking the Automatic
Ant, David Gale
discusses three families of recursively defined sequences of numbers, all
due to Dana Scott and inspired by the Somos sequences:
Sequence 1. Fix a pos... | Yes, we can. The argument for odd $k$ made in the Alman/Cuenca/Huang paper was a red herring. We can argue for arbitrary $k \geq 2$ as follows:
Let $n \geq k+2$. Then, the recursive definition of Sequence 3 yields
\begin{align*}
a_{n}=\dfrac{a_{n-1}a_{n-2}+a_{n-2}a_{n-3}+\cdots +a_{n-k+2}a_{n-k+1}}{a_{n-k}}
\end{align*... | {
"language": "en",
"url": "https://mathoverflow.net/questions/378121",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Are all integers not congruent to 6 modulo 9 of the form $x^3+y^3+3z^2$? Are all integers not congruent to 6 modulo 9 of the form $x^3+y^3+3z^2$
for possibly negative integers $x,y,z$?
We have the identity $ (-t)^3+(t-1)^3+3 t^2=3t-1$.
The only congruence obstruction we found is 6 modulo 9.
| We can also say the each $n \in \mathbb{Z}$ with $n \equiv 3 \pmod 9$ is representable as $x^3 + y^3 + 3z^2$. This is because
$$(-t)^3 + (t-9)^3 + 3(3t -13)^2 = 9t - 222$$
and $-222 \equiv 3 \pmod 9$.
So, along with the identity in the question we can represent each integer congruent to $2$, $3$, $5$, or $8 \pmod 9$. T... | {
"language": "en",
"url": "https://mathoverflow.net/questions/378968",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 0
} |
Explicit eigenvalues of matrix? Consider the matrix-valued operator
$$A = \begin{pmatrix} x & -\partial_x \\ \partial_x & -x \end{pmatrix}.$$
I am wondering if one can explicitly compute the eigenfunctions of that object on the space $L^2(\mathbb R)$?
| First some heuristics, before constructing the complete answer -
this looks a bit more transparent if one considers
$$
A^2 = \begin{pmatrix} -\partial_{x}^{2} +x^2 & 1 \\ 1 & -\partial_{x}^{2} +x^2 \end{pmatrix}
$$
Then, denoting the standard harmonic oscillator eigenfunctions (i.e., the eigenfunctions of $-\partial_{x... | {
"language": "en",
"url": "https://mathoverflow.net/questions/403523",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Sequence of $k^2$ and $2k^2$ ordered in ascending order Let $\eta(n)$ be A006337, an "eta-sequence" defined as follows:
$$\eta(n)=\left\lfloor(n+1)\sqrt{2}\right\rfloor-\left\lfloor n\sqrt{2}\right\rfloor$$
Sequence begins
$$1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1$$
Let $a(n)$ be A091524, $a(m)... | Denote by $f(n)$ the sequence of squares and double squares in ascending order. We have to prove that $f(n)=b(n)=(a(n))^2\eta(n)$. Consider two cases.
*
*$f(n)=k^2$. Then the number of squares and double squares not exceeding $k^2$ equals $n$, that is, $n=k+\lfloor k/\sqrt{2}\rfloor$. Therefore $n<k(1+1/\sqrt{2})$ t... | {
"language": "en",
"url": "https://mathoverflow.net/questions/410799",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Integer solutions of an algebraic equation I'm trying to find integer solutions $(a,b,c)$ of the following algebraic equation with additional conditions $b>a>0$, $c>0$.
$(-a^2+b^2+c^2)(a^2-b^2+c^2)(a^2+b^2-c^2) + 2 a b (-a^2+b^2+c^2)(a^2-b^2+c^2) - 2 b c (a^2-b^2+c^2)(a^2+b^2-c^2) - 2 a c (-a^2+b^2+c^2) (a^2+b^2-c^2) =... | The equation is homogeneous in 3 variables, thus it is associated with a plane curve. First, I would check if the curve has genus less than 2. If the genus is 0 or 1, the curve is parametrizable or elliptic, respectively. In particular, for parametrizable curves, you can generate integer solutions, provided that at lea... | {
"language": "en",
"url": "https://mathoverflow.net/questions/438582",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
Which Diophantine equations can be solved using continued fractions? Pell equations can be solved using continued fractions. I have heard that some elliptic curves can be "solved" using continued fractions. Is this true?
Which Diophantine equations other than Pell equations can be solved for rational or integer points ... | I should have stuck with your preferred notation, as in your $B^2 + B C - 57 C^2 = A^3$ in a comment. So the form of interest will be $x^2 + x y - 57 y^2.$The other classes with this discriminant of indefinite integral binary quadratic forms would then be given by
$ 3 x^2 \pm xy - 19 y^2.$
Therefore, take
$$ \phi(x,y... | {
"language": "en",
"url": "https://mathoverflow.net/questions/77986",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 4,
"answer_id": 3
} |
Fermat's proof for $x^3-y^2=2$ Fermat proved that $x^3-y^2=2$ has only one solution $(x,y)=(3,5)$.
After some search, I only found proofs using factorization over the ring $Z[\sqrt{-2}]$.
My question is:
Is this Fermat's original proof? If not, where can I find it?
Thank you for viewing.
Note: I am not expecting to fin... | Lemma.
Let $a$ and $b$ be coprime integers, and let $m$ and $n$ be positive integers such that $a^2+2b^2=mn$. Then there are coprime integers $r$ and $s$ such that $m=r^2+2s^2$ divides $br-as$. Furthermore, for any such choice of $r$ and $s$, there are coprime integers $t$ and $u$ such that $a=rt-2su$, $b=ru+st$, and $... | {
"language": "en",
"url": "https://mathoverflow.net/questions/142220",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "30",
"answer_count": 5,
"answer_id": 0
} |
Is this a new formula? $\Delta^d x^n/d! = \sum_k \left[ x \atop k\right]{ k+n \brace x + d}(-1)^{x+k}$
$$\frac{\Delta^d x^n}{d!} = \sum_k \left[ x \atop k\right]{ k+n \brace x + d}(-1)^{x+k}$$
Where $x$, $n$ and $d$ are non-negative integers, $\Delta^d$ is the $d$-th difference with respect to $x$, $\left[ x \atop k ... | It is well known that the differencing operator $\Delta$ behaves nicely in the polynomial basis given by the falling factorials: $\Delta(x^{\underline{k}}) = kx^{\underline{k-1}}$. It's also well known that the Stirling numbers are the coefficients that arise when changing from the basis $\{x^k\}$ to the basis $\{x^{\u... | {
"language": "en",
"url": "https://mathoverflow.net/questions/161830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
} |
Primes dividing $2^a+2^b-1$ From Fermat's little theorem we know that every odd prime $p$ divides $2^a-1$ with $a=p-1$.
Is it possible to prove that there are infinitely many primes not
dividing $2^a+2^b-1$?
(With $2^a,2^b$ being incoguent modulo $p$)
1. Obviously, If $2$ is not a quadratic residue modulo $p$ th... | This is a heuristic which suggests that the problem is probably quite hard. We have that $p | 2^{a} + 2^{b} - 1$ if and only if there is some integer $k$, $1 \leq k \leq p-1$ with $k \ne \frac{p+1}{2}$ for which $2^{a} \equiv k \pmod{p}$ and $2^{b} \equiv 1-k \pmod{p}$ are both solvable. If $r$ is the order of $2$ modu... | {
"language": "en",
"url": "https://mathoverflow.net/questions/172706",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 1,
"answer_id": 0
} |
A divisor sum congruence for 8n+6 Letting $d(m)$ be the number of divisors of $m$, is it the case that for $m=8n+6$,
$$ d(m) \equiv \sum_{k=1}^{m-1} d(k) d(m-k) \pmod{8}\ ?$$
It's easy to show that both sides are 0 mod 4: the left side since two primes appear to odd order in the factorization of $m$, and the right side... | The congruence you state is true for all $m \equiv 6 \pmod{8}$. The proof I give below relies on the theory of modular forms. First, observe that
$$
\sum_{k=1}^{m-1} d(k) d(m-k) = 2 \sum_{k=1}^{\frac{m-2}{2}} d(k) d(m-k) + d\left(\frac{m}{2}\right)^{2}.
$$
Noting that $d(m) \equiv d\left(\frac{m}{2}\right)^{2} \pmod{8}... | {
"language": "en",
"url": "https://mathoverflow.net/questions/177477",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
} |
Does the Lehmer quintic parameterize certain minimal polynomials of the $p$th root of unity for infinitely many $p$? The solvable Emma Lehmer quintic is given by,
$$F(y) = y^5 + n^2y^4 - (2n^3 + 6n^2 + 10n + 10)y^3 + (n^4 + 5n^3 + 11n^2 + 15n + 5)y^2 + (n^3 + 4n^2 + 10n + 10)y + 1 = 0$$
with discriminant $D = (7 + 10 n... | For question 1, the answer is yes, as shown by Emma Lehmer herself. (See the paper here, in particular, equation (5.8) on page 539.) In particular, Lehmer states that one can take
$$
a = \frac{\left(\frac{n}{5}\right) - n^{2}}{5}, \quad b = \left(\frac{n}{5}\right).
$$
(Here $\left(\frac{n}{5}\right)$ denotes the Le... | {
"language": "en",
"url": "https://mathoverflow.net/questions/190893",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How do we show this matrix has full rank?
I met with the following difficulty reading the paper Li, Rong Xiu "The properties of a matrix order column" (1988):
Define the matrix $A=(a_{jk})_{n\times n}$, where
$$a_{jk}=\begin{cases}
j+k\cdot i&j<k\\
k+j\cdot i&j>k\\
2(j+k\cdot i)& j=k
\end{cases}$$
and $i^2=-1$.
... | I use Christian Remling idea,In fact,I can find the matrix $$B_{ij}=\min{\{i,j\}}$$eigenvalue is
$$\dfrac{1}{4\sin^2{\dfrac{j\pi}{2(n+1)}}},j=1,2,\cdots,n$$
proof:
then we have
$$B=\begin{bmatrix}
1&1&1&\ddots&1&1\\
1&2&2&\ddots&\ddots&2\\
1&2&3&3&\ddots&3\\
\vdots&\ddots&\ddots&\ddots&\ddots&\cdots\\
1&\vdots&\ddot... | {
"language": "en",
"url": "https://mathoverflow.net/questions/191796",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 3,
"answer_id": 0
} |
proving that a smooth curve in Euclidean n-space contains n+1 affinely independent points If I let $f(\theta)=((\mathrm{cos} \theta)X+(\mathrm{sin} \theta)Y)^{n-1}$ and view the range of this curve as a subset of the space of homogeneous polynomials of degree $n-1$ in two variables viewed as an $n$-dimensional Euclidea... | This isn't true if $n$ is odd. For example, if $n=3$, then your formula is $(a,b,c) = (\cos^2 \theta, 2 \sin \theta \cos \theta, \sin^2 \theta)$ and it always lies in the hyperplane $a+c=1$. More generally, whenever $n$ is odd, the equality $1 = (\sin^2 \theta+ \cos^2 \theta)^{(n-1)/2} = \sum \binom{(n-1)/2}{k} \sin^{2... | {
"language": "en",
"url": "https://mathoverflow.net/questions/199453",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find all solution $a,b,c$ with $(1-a^2)(1-b^2)(1-c^2)=8abc$ Two years ago, I made a conjecture on stackexchange:
Today, I tried to find all solutions in integers $a,b,c$ to
$$(1-a^2)(1-b^2)(1-c^2)=8abc,\quad a,b,c\in \mathbb{Q}^{+}.$$
I have found some solutions, such as
$$(a,b,c)=(5/17,1/11,8/9),(1/7,5/16,9/11),(3/4... | I'm late for this party, but using math110's method employing Euler bricks, couldn't resist giving some simple rational solutions to,
$$(1-a^2)(1-b^2)(1-c^2) = 8abc$$
Solution 1:
$$a,\,b,\,c = \frac{-(x-z)(2x+z)}{(2x-z)y},\;\frac{z}{2x},\;\frac{-2y+z}{2y+z}\tag1$$
where $x^2+y^2=z^2.$
Solution 2:
$$a,\,b,\,c = \frac{2z... | {
"language": "en",
"url": "https://mathoverflow.net/questions/208485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 4,
"answer_id": 1
} |
Is it possible that $(a,b,c)$, $(x,y,a)$, $(p,q,b)$ are Pythagorean triples simultaneously? Do there exist postive integers $a,b,c,x,y,p,q$ such $(a,b,c)$, $(x,y,a)$, $(p,q,b)$ are all Pythagorean triples? That is, does the system
$$\begin{cases}
a^2+b^2=c^2\\
x^2+y^2=a^2\\
p^2+q^2=b^2
\end{cases}$$
have a postive inte... | The system of equations.
$$\left\{\begin{aligned}&a^2+b^2=c^2\\&x^2+y^2=a^2\\&f^2+q^2=b^2\end{aligned}\right.$$
Equivalent to the need to solve the following system of equations.
$$\left\{\begin{aligned}&a=2ps=z^2+t^2\\&b=p^2-s^2=j^2+v^2\\&c=p^2+s^2\\&x=2zt\\&y=z^2-t^2\\&f=2jv\\&q=j^2-v^2\end{aligned}\right.$$
To ease ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/208891",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
An inequality improvement on AMM 11145 I have asked the same question in math.stackexchange, I am reposting it here, looking for answers:
How to show that for $a_1,a_2,\cdots,a_n >0$ real numbers and for $n \ge 3$:
$$\sum_{k=1}^{n}\dfrac{k}{a_{1}+a_{2}+\cdots+a_{k}}\le\left(2-\dfrac{7\ln{2}}{8\ln{n}}\right)\sum_{k=1}^... | I came up with something years ago which is similar to Fedor Petrov's (or other users').
Problem: Let $a_i > 0; \ i = 1, 2, \cdots, n$ ($n\ge 2$). Let $C(n) = 2 - \frac{7\ln 2}{8\ln n}$. Prove that
$$\sum_{k=1}^n \frac{k}{a_1 + a_2 + \cdots + a_k}
\le C(n)\sum_{k=1}^n \frac{1}{a_k}.$$
Introducing the coefficients (to ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/224616",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 3,
"answer_id": 2
} |
Investigation of $\sum \limits_{k=-\infty}^\infty \frac{x^{k+n}}{ \Gamma(k+n+1)}$ where $n \in C$? $$e^x=\sum \limits_{k=0}^\infty \frac{x^k}{k!}$$
We can rewrite the equation as
$$e^x=\sum \limits_{k=0}^\infty \frac{x^k}{ \Gamma(k+1)} \tag{1}$$
because $x!=\Gamma(x+1)$ where $x$ is non-negative integer.
$\Gamma(x)$ (... | You might like to look at asymptotic expansions--a rich, fascinating field. See this excerpt for a quick intro and the book by Dingle, available courtesy of Michael Berry, who has himself written many fine articles on the subject, e.g., this one. (Wikipedia has a short bibliography.)
Coincidentally, just last night I w... | {
"language": "en",
"url": "https://mathoverflow.net/questions/227642",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Tricky two-dimensional recurrence relation I would like to obtain a closed form for the recurrence relation
$$a_{0,0} = 1,~~~~a_{0,m+1} = 0\\a_{n+1,0} = 2 + \frac 1 2 \cdot(a_{n,0} + a_{n,1})\\a_{n+1,m+1} = \frac 1 2 \cdot (a_{n,m} + a_{n,m+2}).$$
Even obtaining a generating function for that seems challenging. Is ther... | Here is the table for $a_{n,m}2^m$ (since these are integers):
$$\begin{pmatrix}
1 & 5 & 14 & 35 & 82 & 186 & 412 & 899 & 1938 \\
0 & 1 & 5 & 15 & 40 & 98 & 231 & 527 & 1180 \\
0 & 0 & 1 & 5 & 16 & 45 & 115 & 281 & 660 \\
0 & 0 & 0 & 1 & 5 & 17 & 50 & 133 & 336 \\
0 & 0 & 0 & 0 & 1 & 5 & 18 & 55 & 152 \\
0 & 0 & 0... | {
"language": "en",
"url": "https://mathoverflow.net/questions/235041",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
Rogers-Ramanujan continued fraction $R(e^{-2 \pi \sqrt 5})$ Let $$R(q) = \cfrac{q^{1/5}}{1 + \cfrac{q}{1 + \cfrac{q^{2}}{1 + \cfrac{q^{3}}{1 + \cdots}}}}$$
It is easy to evaluate $R(e^{-2 \pi/ \sqrt 5})$ using the Dedekind eta function identity $\eta(-\frac{1}{z})=\sqrt{-iz}\eta(z)$
and one of the most fundamental... | Let $R(q)$ be the Rogers-Ramanujan continued fraction
$$
R(q):=\frac{q^{1/5}}{1+}\frac{q^1}{1+}\frac{q^2}{1+}\frac{q^3}{1+}\ldots,|q|<1
$$
Let also for $r>0$
$$
Y=Y(r):=R(e^{-2\pi\sqrt{r}})^{-5}-11-R(e^{-2\pi\sqrt{r}})^5
$$
It is easy to show someone that
$$
Y\left(\frac{r}{5}\right)Y\left(\frac{1}{5r}\right)=125, : (... | {
"language": "en",
"url": "https://mathoverflow.net/questions/241809",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
} |
Is $x^{2k+1} - 7x^2 + 1$ irreducible?
Question. Is the polynomial $x^{2k+1} - 7x^2 + 1$ irreducible over $\mathbb{Q}$ for every positive integer $k$?
It is irreducible for all positive integers $k \leq 800$.
| Here is a proof, based on a trick that can be used to prove that
$x^n + x + 1$ is irreducible when $n \not\equiv 2 \bmod 3$.
We work with Laurent polynomials in $R = \mathbb Z[x,x^{-1}]$; note that
$R$ has unit group $R^\times = \pm x^{\mathbb Z}$.
We observe that for $f \in R$, the sum of the squares of the coefficien... | {
"language": "en",
"url": "https://mathoverflow.net/questions/258914",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "28",
"answer_count": 3,
"answer_id": 0
} |
Evaluation of a double definite integral with a singularity How to compute the
$$\int_{0}^{1} \int_{0}^{1} \frac{(\log(1+x^2)-\log(1+y^2))^2 }{|x-y|^{2}}dx dy.$$
Is it possible to compute the integral analytically upto some terms. I believe it should involve hypergeometric series. Any ideas are welcome.
| With some effort, Mathematica evaluates this as
$$\int_{0}^{1} \int_{0}^{1} \frac{[\ln(1+x^2)-\ln(1+y^2)]^2 }{(x-y)^{2}}dx dy=2 \sqrt{2} \; _4F_3\left(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2};\tfrac{3}{2},\tfrac{3}{2},\tfrac{3}{2};\tfrac{1}{2}\right)+(4 \pi-\ln 2) C-2 i \,\text{Li}_3\left(\tfrac{1}{2}+\tfra... | {
"language": "en",
"url": "https://mathoverflow.net/questions/327767",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Does this degree 12 genus 1 curve have only one point over infinitely many finite fields? Let $F(x,y,z)$ be the degree 12 homogeneous polynomial:
$$x^{12} - x^9 y^3 + x^6 y^6 - x^3 y^9 + y^{12} - 4 x^9 z^3 + 3 x^6 y^3 z^3 - 2 x^3 y^6 z^3 + y^9 z^3 + 6 x^6 z^6 - 3 x^3 y^3 z^6 + y^6 z^6 - 4 x^3 z^9 + y^3 z^9 + z^{12}$$
O... | The polynomial you wrote is the product of the four polynomials $x^3 - r y^3 - z^3$, where $r$ is a root of the polynomial $t^4 - t^3 + t^2 - t + 1$. I did not read your reference, but likely they assume that the curves are geometrically irreducible.
| {
"language": "en",
"url": "https://mathoverflow.net/questions/332515",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Polynomial inequality of sixth degree There is the following problem.
Let $a$, $b$ and $c$ be real numbers such that $\prod\limits_{cyc}(a+b)\neq0$ and $k\geq2$ such that $\sum\limits_{cyc}(a^2+kab)\geq0.$
Prove that:
$$\sum_{cyc}\frac{2a^2+bc}{(b+c)^2}\geq\frac{9}{4}.$$
I have a proof of this inequality for any... | We want to show that your inequality does not hold for $k\in[2,13/5)$. In view of the identity in your answer, it is enough to show that for each $k\in[2,13/5)$ there is a triple $(a,b,c)\in\mathbb R^3$ with the following properties: $a=-1>b$,
\begin{align}s_4&:=\sum_{cyc}(2a^3-a^2b-a^2c) \\
&=a^2 (2 a-b-c)+b^2 (-a+2 ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/358054",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 2,
"answer_id": 1
} |
Maximum eigenvalue of a covariance matrix of Brownian motion $$ A := \begin{pmatrix}
1 & \frac{1}{2} & \frac{1}{3} & \cdots & \frac{1}{n}\\
\frac{1}{2} & \frac{1}{2} & \frac{1}{3} & \cdots & \frac{1}{n}\\
\frac{1}{3} & \frac{1}{3} & \frac{1}{3} & \cdots & \frac{1}{n}\\
\vdots & \vdots & \vdots & \ddots & \frac{1}{n}\\
... | Inspired by @Mateusz Wasilewski I find another method.
\begin{eqnarray*}
\langle x,Ax\rangle & = & \langle Lx,Lx\rangle\\
& = & \sum_{i=1}^{n}u_{i}^{2}
\end{eqnarray*}
where $u_{i}=\sum_{j=i}^{n}\frac{1}{j}x_{j}$.
\begin{eqnarray*}
\sum_{i=1}^{n}u_{i}^{2} & = & \sum_{i=1}^{n}(\sum_{k=i}^{n}b_{k})^{2}\quad(\text{where}... | {
"language": "en",
"url": "https://mathoverflow.net/questions/366339",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Are all integers not congruent to 6 modulo 9 of the form $x^3+y^3+3z^2$? Are all integers not congruent to 6 modulo 9 of the form $x^3+y^3+3z^2$
for possibly negative integers $x,y,z$?
We have the identity $ (-t)^3+(t-1)^3+3 t^2=3t-1$.
The only congruence obstruction we found is 6 modulo 9.
| We found another approach based on integral points on genus 0 curves.
Let $G(x,z,a_1,a_2,a_3)=(a_1 x+a_2)^3+(-a_1 x+a_3)^3+3 z^2$.
For fixed $n$ and $a_i$, $G=n$ is degree two genus 0 curve and it might
have infinitely many integral points, which gives infinitely many solutions
to the OP.
WolramAlpna on solve (10x-5)^3... | {
"language": "en",
"url": "https://mathoverflow.net/questions/378968",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 1
} |
On some inequality (upper bound) on a function of two variables There is a problem (of physical origin) which needs an analytical solution or a hint. Let us consider the following real-valued function of two variables
$y (t,a) = 4 \left(1 + \frac{t}{x(t,a)}\right)^{ - a - 1/2}
\left(1 - \frac{2}{x(t,a)}\right)^{ 1... | a human verifiable proof:
Let us prove that the sharp bound of $y$ is $\frac{\sqrt{3+2\sqrt 3}}{3}$.
Letting $x := x(t, a), z := z(t, a), y := y(t, a)$, we have
$$y = 4 \left(\frac{1 + t/2}{1 + t/x}\right)^a\cdot \left(\frac{z}{(1 + t/x)^2}\right)^{1/4} (1 - 2/x)^{1/2} x^{-3/2}. \tag{1}$$
Using Bernoulli inequality $(1... | {
"language": "en",
"url": "https://mathoverflow.net/questions/384678",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
A tricky integral to evaluate I came across this integral in some work. So, I would like to ask:
QUESTION. Can you evaluate this integral with proofs?
$$\int_0^1\frac{\log x\cdot\log(x+2)}{x+1}\,dx.$$
| $$\int_0^1\frac{\ln x\cdot\ln(x+2)}{x+1}\,dx=$$
$$=\text{Li}_3\left(-\tfrac{1}{3}\right)-2 \,\text{Li}_3\left(\tfrac{1}{3}\right)+\tfrac{1}{2} \ln 3\left[ \text{Li}_2\left(\tfrac{1}{9}\right)-6\, \text{Li}_2\left(\tfrac{1}{3}\right) -\tfrac{2}{3} \ln ^2 3\right]+\tfrac{13}{8} \zeta (3).$$
I checked that this combinati... | {
"language": "en",
"url": "https://mathoverflow.net/questions/385258",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 0
} |
Improper integral $\int_0^\infty\frac{x^{2n+1}}{(1+x^2)^2(2+x^2)^N}dx,\ \ \ n\le N$ How can I evaluate this integral?
$$\int_0^\infty\frac{x^{2n+1}}{(1+x^2)^2(2+x^2)^N}dx,\ \ \ n\le N$$
Maybe there is a recurrence relation for the integral?
| Let $I_{n,N}$ denote the integral in question, where $n$ and $N$ are nonnegative integers such that $n\le N$. With the change of variables $y=2+x^2$ and then using the binomial expansion of $(y-2)^n$, we get
\begin{equation}
2I_{n,N}=\int_2^\infty\frac{(y-2)^n\,dy}{(y-1)^2y^N}
=\sum_{j=0}^n\binom nj (-2)^{n-j}J... | {
"language": "en",
"url": "https://mathoverflow.net/questions/393753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 0
} |
The constant $\pi$ expressed by an infinite series I am looking for the proof of the following claim:
First, define the function $\operatorname{sgn_1}(n)$ as follows:
$$\operatorname{sgn_1}(n)=\begin{cases} -1 \quad \text{if } n \neq 3 \text{ and } n \equiv 3 \pmod{4}\\1 \quad \text{if } n \in \{2,3\} \text{ or } n \eq... | This can be proved similarly as Alexander Kalmynin's method .
Let, the sum be $S$, then we can make the following identity because $\text{sgn}_1$ of $2,3$ is defined to be $1$. So, $\text{sgn}_1(ak)=\text{sgn}_1(k), a=2,3,6$. Also, from the definition of $\text{sgn}_2$ we can see, $\text{sgn}_2(n)=1,-1$ respectively fo... | {
"language": "en",
"url": "https://mathoverflow.net/questions/395438",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Integrality of a sequence formed by sums Consider the following sequence defined as a sum
$$a_n=\sum_{k=0}^{n-1}\frac{3^{3n-3k-1}\,(7k+8)\,(3k+1)!}{2^{2n-2k}\,k!\,(2k+3)!}.$$
QUESTION. For $n\geq1$, is the sequence of rational numbers $a_n$ always integral?
| Let $A(x) = \sum_{n=1}^\infty a_n x^n$ and let
$$S(x) = \sum_{k=0}^\infty (7k+8)\frac{(3k+1)!}{k!\,(2k+3)!} x^k.$$
Then the formula for $a_n$ gives
$A(x) = R(x)S(x)$,
where
$$R(x) = \frac{1}{3}\biggl(\frac{1}{1-\frac{27}{4} x} -1\biggr).$$
A standard argument, for example by Lagrange inversion, gives
$$S\left(\frac{y}{... | {
"language": "en",
"url": "https://mathoverflow.net/questions/398037",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 4,
"answer_id": 2
} |
Subsequences of odd powers Let $p$ and $q$ be integers.
Let $f(n)$ be A007814, the exponent of the highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $n$.
Then we have an integer sequence given by
\begin{align}
a(0)=a(1)&=1\\
a(2n)& = pa(n)+qa(2n-2^{f(... | Quite similarly to my answer to the previous question, we have that for $n=2^tk$ with odd $k$,
$$
a(n)=\sum_{i=0}^t \binom{t}{i}p^{t-i}q^i a(2^i(k-1)+1).
$$
It further follows that for $n=2^{t_1}(1+2^{t_2+1}(1+\dots(1+2^{t_s+1}))\dots)$ with $t_j\geq 0$, we have
\begin{split}
a(n) &= \sum_{i_1=0}^{t_1} \binom{t_1}{i_1}... | {
"language": "en",
"url": "https://mathoverflow.net/questions/405969",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How big can a triangle be, whose sides are the perpendiculars to the sides of a triangle from the vertices of its Morley triangle? Given any triangle $\varDelta$, the perpendiculars from the vertices of its (primary) Morley triangle to their respective (nearest) side of $\varDelta$ intersect in a triangle $\varDelta'$,... | We will perform computation using
normed barycentric coordinates in the given triangle $\Delta ABC$ with side lengths $a,b,c$ and angles
$\hat A=3\alpha$,
$\hat B=3\beta$,
$\hat C=3\gamma$.
We denote by $R$ the circumradius of the circle $\odot(ABC)$, and the area $[ABC]$ by $S$.
Trilinear coordinates of the points $A'... | {
"language": "en",
"url": "https://mathoverflow.net/questions/417175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Integral of a product between two normal distributions and a monomial The integral of the product of two normal distribution densities can be exactly solved, as shown here for example.
I'm interested in compute the following integral (for a generic $n \in \mathbb{N}$):
$$
I_n = \int_{-\infty}^{\infty} \frac{1}{\sqrt{2 ... | $\def\m{\mu}
\def\p{\pi}
\def\s{\sigma}
\def\f{\varphi}
\def\r{\rho}
\def\mm{M}
\def\ss{S}$Let
\begin{align*}
\f(x;\m,\s) &= \frac{1}{\sqrt{2\p}\s} e^{-(x-\m)^2/(2\s^2)}
\end{align*}
and
\begin{align*}
\f_m(x) &= \prod_{i=1}^m \f(x;\m_i,\s_i) \\
&= \frac{1}{(2\p)^{m/2}\prod_{i=1}^m \s_i}
\exp\left(-\sum_{i=1}^m \frac{(... | {
"language": "en",
"url": "https://mathoverflow.net/questions/437171",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Integration of the product of pdf & cdf of normal distribution Denote the pdf of normal distribution as $\phi(x)$ and cdf as $\Phi(x)$. Does anyone know how to calculate $\int \phi(x) \Phi(\frac{x -b}{a}) dx$? Notice that when $a = 1$ and $b = 0$ the answer is $1/2\Phi(x)^2$. Thank you!
| We have $\phi(x)=\frac 1{\sqrt{2\pi}}\exp\left(-\frac{†^2}2\right)$ and $\Phi(x)=\int_{-\infty}^x\phi(t)dt$. We try to compute
$$ I(a,b):=\int\phi(x)\Phi\left(\frac{x-b}a\right)dx.$$
Using the dominated convergence theorem, we are allowed to take the derivative with respect to $b$ inside the integral. We have
$$\parti... | {
"language": "en",
"url": "https://mathoverflow.net/questions/101469",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 0
} |
Logarithm of the hypergeometric function For $F(x)={}_2F_1 (a,b;c;x)$, with $c=a+b$, $a>0$, $b>0$, it has been proved in [1] that $\log F(x)$ is convex on $(0,1)$.
I numerically checked that with a variety of $a,\ b$ values, $\log F(x)$ is not only convex, but also has a Taylor series in x consisting of strictly positi... | Here's a sketch of a proof of a stronger statement: the coefficients of the Taylor series for $\log{}_2F_1(a,b;a+b+c;x)$ are rational functions of $a$, $b$, and $c$ with positive coefficients.
To see this we first note that
$$\begin{aligned}
\frac{d\ }{dx} \log {}_2F_1(a,b;a+b+c;x) &=
\frac{\displaystyle
\frac{d\ }{d... | {
"language": "en",
"url": "https://mathoverflow.net/questions/143350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
How many solutions to $2^a + 3^b = 2^c + 3^d$? A few weeks ago, I asked on math.stackexchange.com how many quadruples of non-negative integers $(a,b,c,d)$ satisfy the following equation:
$$2^a + 3^b = 2^c + 3^d \quad (a \neq c)$$
I found 5 quadruples: $5 = 2^2 + 3^0 = 2^1 + 3^1$, $11 = 2^3 + 3^1 = 2^1 + 3^2$, $17 = 2... | The answer is yes, there are finitely many for positive $a, b, c, d$, and you've found all of them.
See Theorem 7 of R. Scott, R. Styer, On the generalized Pillai equation $\pm a^x \pm b^y = c$, Journal of Number Theory, 118 (2006), 236–265. I quote:
Theorem 7. Let $a$ be prime, $a>b$, $b = 2$ or $b = 3$, $a$ not a l... | {
"language": "en",
"url": "https://mathoverflow.net/questions/164624",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 6,
"answer_id": 4
} |
How do we show this matrix has full rank?
I met with the following difficulty reading the paper Li, Rong Xiu "The properties of a matrix order column" (1988):
Define the matrix $A=(a_{jk})_{n\times n}$, where
$$a_{jk}=\begin{cases}
j+k\cdot i&j<k\\
k+j\cdot i&j>k\\
2(j+k\cdot i)& j=k
\end{cases}$$
and $i^2=-1$.
... | OK, let me try again, maybe I'll get it right this time. I'll show that $P$ is positive definite. This will imply the claim because if $(P+iQ)(x+iy)=0$ with $x,y\in\mathbb R^n$, then $Px=Qy$, $Py=-Qx$, and by taking scalar products with $x$ and $y$, respectively, we see that $\langle x, Px \rangle = -\langle y, Py\rang... | {
"language": "en",
"url": "https://mathoverflow.net/questions/191796",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 3,
"answer_id": 1
} |
Angle subtended by the shortest segment that bisects the area of a convex polygon Let $C$ be a convex polygon in the plane and let $s$ be the shortest line segment (I believe this is called a "chord") that divides the area of $C$ in half. What is the smallest angle that $s$ could make where it touches the boundary of $... | I'll prove that for any triangle below, $\displaystyle \theta \geqq \pi /3\ $
Assuming $\displaystyle X'Y' =z$ be the line segment that bisects area $\displaystyle A$ of triangle $\displaystyle XYZ$ .
Let $\displaystyle \angle Z=\alpha $ , $\displaystyle Y'Z=x$ and $\displaystyle X'Z=y$
By Heron's formula, $\displays... | {
"language": "en",
"url": "https://mathoverflow.net/questions/201181",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
} |
Curious inequality Set
$$
g(x)=\sum_{k=0}^{\infty}\frac{1}{x^{2k+1}+1} \quad \text{for} \quad x>1.
$$
Is it true that
$$
\frac{x^{2}+1}{x(x^{2}-1)}+\frac{g'(x)}{g(x)}>0 \quad \text{for}\quad x>1?
$$
The answer seems to be positive. I spent several hours in proving this statement but I did not come up with anythi... | The inequality is equivalent to
$$S := (x^2+1)g(x) + x(x^2-1)g'(x) > 0.$$
The left hand side here can be expanded to
$$S = \sum_{k\geq 0} \frac{(x^2+1)(x^{2k+1}+1) - (2k+1)x^{2k+1}(x^2-1)}{(x^{2k+1}+1)^2} $$
$$= \sum_{k\geq 0} \frac{(x^2+1) - (2k+1)(x^2-1)}{x^{2k+1}+1} + \sum_{k\geq 0}\frac{(2k+1)(x^2-1)}{(x^{2k+1}+1... | {
"language": "en",
"url": "https://mathoverflow.net/questions/217711",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
"answer_count": 3,
"answer_id": 2
} |
Are the following identities well known? $$
x \cdot y = \frac{1}{2 \cdot 2 !} \left( (x + y)^2 - (x - y)^2 \right)
$$
$$
\begin{eqnarray}
x \cdot y \cdot z &=& \frac{1}{2^2 \cdot 3 !} ((x + y + z)^3 - (x + y - z)^3 \nonumber \\
&-& (x - y + z)^3 + (x - y - z)^3 ),
\end{eqnarray}
$$
$$
\begin{eqnarray}
x \cdot y \cdot ... | Although not the exactly the same due to $2^{n-1}$ instead of $2^n$ terms, the OP's formula seems to be essentially the well-known polarization formula for homogeneous polynomials, which is stated as following:
Any polynomial $f$, homogeneous of degree $n$ can be written as $f(x)=H(x,\ldots,x)$ for a specific multilin... | {
"language": "en",
"url": "https://mathoverflow.net/questions/220447",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 1,
"answer_id": 0
} |
Why is this not a perfect square? Let $x, y$ and $z$ be positive integers with $x<y$. It appears that the integer
$$(y^2z^3-x)(y^2z^3+3x)$$
is never a perfect square. Why? A proof? I'm not sure if it is easy.
| Rewrite your expression as $(y^2z^3+x)^2-4x^2$. This is clearly less than $(y^2z^3+x)^2$. On the other hand, it is larger than $(y^2z^3+x-2)^2=(y^2z^3+x)^2-4(y^2z^3+x)+4$, since $4(y^2z^3+x)-4\geq 4y^2+4x-4\geq 4y^2>4x^2$. So if this expression is a square, it must be $(y^2z^3+x-1)^2=(y^2z^3+x)^2-2(y^2z^3+x)+1$, so $4x... | {
"language": "en",
"url": "https://mathoverflow.net/questions/250163",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Algorithm to decide whether two constructible numbers are equal? The set of constructible numbers
https://en.wikipedia.org/wiki/Constructible_number
is the smallest field extension of $\mathbb{Q}$ that is closed under square root and complex conjugation. I am looking for an algorithm that decides if two constructible ... | Although this can be done using the complicated algorithms for general algebraic numbers, there’s a much simpler recursive algorithm for constructible numbers that I implemented in the Haskell constructible library.
A constructible field extension is either $\mathbb Q$ or $F{\left[\sqrt r\right]}$ for some simpler cons... | {
"language": "en",
"url": "https://mathoverflow.net/questions/264827",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Existence of generalized inverse-like operator Does there exist an operator, $\star$, such that for all full rank matrices $B$ and all $A$ of appropriate dimensions:
$$
B(B^\intercal AB)^\star B^\intercal = A^\star,
$$
and such that $A^\star=0$ if and only if $A=0$?
Edit: Also, $\star : \operatorname{M}(m,n,\mathbb R) ... | No. We suppose it is defined for $2\times 2$ matrices and we get a contradiction. Unless you drop the requirement on $\operatorname{rank} A^\star$ in which case $A^\star=0$ trivially works.
Let $A=\begin{pmatrix} 1 & 0 \\ 0 & 0\end{pmatrix}$ and let $B=\begin{pmatrix} a & b\\ c & d\end{pmatrix}$ with arbitrary $a,b,c,d... | {
"language": "en",
"url": "https://mathoverflow.net/questions/270049",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Matrix rescaling increases lowest eigenvalue? Consider the set $\mathbf{N}:=\left\{1,2,....,N \right\}$ and let $$\mathbf M:=\left\{ M_i; M_i \subset \mathbf N \text{ such that } \left\lvert M_i \right\rvert=2 \text{ or }\left\lvert M_i \right\rvert=1 \right\}$$
be the set of all subsets of $\mathbf{N}$ that are of car... | The matrices are of the form
$$
A=\begin{pmatrix} 1 & C \\ C^* & D \end{pmatrix}, \quad\quad
B= \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix} A \begin{pmatrix} 1 & 0\\ 0&2\end{pmatrix} ,
$$
with the blocks corresponding to the sizes of the sets $M_j$ involved.
Let $v=(x,y)^t$ be a normalized eigenvector for the minimum ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/313470",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
How can I simplify this sum any further? Recently I was playing around with some numbers and I stumbled across the following formal power series:
$$\sum_{k=0}^\infty\frac{x^{ak}}{(ak)!}\biggl(\sum_{l=0}^k\binom{ak}{al}\biggr)$$
I was able to "simplify" the above expression for $a=1$:
$$\sum_{k=0}^\infty\frac{x^k}{k!}\c... | You might be able to use the fact that
$$\sum_{k=0}^\infty b_{ak}=\sum_{k=0}^\infty \left(\frac{1}{a}\sum_{j=0}^{a-1} \exp\left(2\pi ijk/a\right)\right)b_k.$$
For example, when $a=1$, taking $b_k = \frac{x^k}{k!}\sum_{\ell \ge 0} \binom{k}{\ell}$ yields
$$\sum_{k=0}^\infty b_{k}=\sum_{k=0}^\infty \frac{x^k}{k!}\sum_{\e... | {
"language": "en",
"url": "https://mathoverflow.net/questions/346198",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 2
} |
Can an even perfect number be a sum of two cubes? A similar question was asked before in https://math.stackexchange.com/questions/2727090/even-perfect-number-that-is-also-a-sum-of-two-cubes, but no conclusions were drawn.
On the Wikipedia article of perfect numbers there are two related results concerning whether an ev... | Here is a proof that 28 is the only even perfect number that is the sum of two positive cubes. The proof in Gallardo's article must be adapted in the case $x,a$ are even.
Write $N=2^{p-1}(2^p-1) = x^3+y^3 = (x+y)(x^2-xy+y^2)$. The gcd $d$ of $x$ and $y$ must be a power of 2, because $d^3$ divides $N$. Writing $x=2^h u$... | {
"language": "en",
"url": "https://mathoverflow.net/questions/375879",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
A special congruence For any $a, b\in\mathbb{N}$ with $a+2b\not\equiv 0\pmod 3$, we define $\delta(a, b)$ as follows:
\begin{align*}
\delta(a, b)={\left\{\begin{array}{rl}
1,\ \ \ \ &{\rm if} \ a+2b\equiv 1\pmod 3,\\
0,\ \ \ \ &{\rm if} \ a+2b\equiv 2\pmod 3.
\end{array}\right.}
\end{align*}
Furthermore, for any $m, n\... | We can rewrite the sum as
$$\sum_{a=0}^{m} (-1)^a \binom{m}{a}\sum_{\substack{b=0 \\ 3 \nmid a+2b}} (-1)^b 2^{\delta(a,b)} \binom{n}{b}$$
*
*Now, when, $a=3k+1$, then we have $b=3k$ or $3k-1$.
Then for, $b=3k$, $\delta(a,b)=1$ and for $b=3k-1$ and $\delta(a,b)=0$.
*Similarly, when $a=3k-1$, $b=3k, \delta(a,b)=0$ a... | {
"language": "en",
"url": "https://mathoverflow.net/questions/380193",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is there a nonzero solution to this infinite system of congruences? Is there a triple of nonzero even integers $(a,b,c)$ that satisfies the following infinite system of congruences?
$$
a+b+c\equiv 0 \pmod{4} \\
a+3b+3c\equiv 0 \pmod{8} \\
3a+5b+9c\equiv 0 \pmod{16} \\
9a+15b+19c\equiv 0 \pmod{32} \\
\vdots \\
s_na + t_... | Let $u_n = a s_n + b t_n + c s_{n+1}$. The stronger claim is true: for large enough values of $n$,
the number $u_n$ will be exactly divisible
by a fixed power of $2$ that doesn't depend on $n$.
Let $u_n = a s_n + b t_n + c s_{n+1}$ then (by induction)
$$u_{n} = u_{n-1} + 2 u_{n-2} + 4 u_{n-3}.$$
The polynomial $x^3 - x... | {
"language": "en",
"url": "https://mathoverflow.net/questions/381057",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
Why $\lim_{n\rightarrow \infty}\frac{F(n,n)}{F(n-1,n-1)} =\frac{9}{8}$? $$F(m,n)= \begin{cases}
1, & \text{if $m n=0$ }; \\
\frac{1}{2} F(m ,n-1) + \frac{1}{3} F(m-1,n )+ \frac{1}{4} F(m-1,n-1), & \text{ if $m n>0$. }%
\end{cases}$$
Please a proof of:
$$\lim_{n\rightarrow \infty}\frac{F(n,n)}{F(n-1,n-1)} =\lim_{n\righ... | We will compute the generating function, and use the method described in section 2 of this paper.
Let $F_{m,n}=F(m,n)$. Consider the generating function
$$G(x,y)=\sum_{m=0}^\infty\sum_{n=0}^\infty F_{m,n}x^my^n.$$
Then the recurrence gives
\begin{align*}
&G(x,y)=\sum_{m=0}^\infty F_{m,0}x^m+\sum_{n=1}^\infty F_{0,n}y^n... | {
"language": "en",
"url": "https://mathoverflow.net/questions/390924",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Improper integral $\int_0^\infty\frac{x^{2n+1}}{(1+x^2)^2(2+x^2)^N}dx,\ \ \ n\le N$ How can I evaluate this integral?
$$\int_0^\infty\frac{x^{2n+1}}{(1+x^2)^2(2+x^2)^N}dx,\ \ \ n\le N$$
Maybe there is a recurrence relation for the integral?
| One approach is to consider the sum
$$ J = \sum_{n,m=0}^\infty s^nt^mI_{n,n+m} = \int_{x=0}^\infty F\,dx, $$
where
$$ F = \sum_{n,m=0}^\infty s^nt^m \frac{x^{2n+1}}{(1+x^2)^2(2+x^2)^{n+m}} =
\frac{x(2+x^2)^2}{(1+x^2)^2(2+x^2-sx^2)(2+x^2-t)}
$$
This works out as
$$
J = \frac{2s^2\ln(1-s)}{(1+s)^2(st-2s-t)} +
\frac... | {
"language": "en",
"url": "https://mathoverflow.net/questions/393753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 1
} |
General formulas for derivative of $f_n(x)=\dfrac{ax^n+bx^{n-1}+cx^{n-2}+\cdots}{a'x^n+b'x^{n-1}+c'x^{n-2}+\cdots},\quad a'\neq0$ For the function $f_1(x)=\dfrac{ax+b}{a'x+b'},\quad a'\neq0$ , we have $$f_1'(x)=\dfrac{\begin{vmatrix}{a} && {b} \\ {a'} && {b'}\end{vmatrix}}{(a'x+b')^2}$$
For $f_2(x)=\dfrac{ax^2+bx+c}{a'... | You can easily extend this, but for $n\geq 3$ you will end up with more than one term per monomial:
For two functions $f$, $g$ rewrite the quotient rule using a determinant
$$\frac{d}{dx} \frac{f}{g} = \frac{\frac{df}{dx}g-f \frac{dg}{dx}}{g^2} = \frac{\begin{vmatrix} \frac{df}{dx} & f \\ \frac{dg}{dx} & g \end{vmatrix... | {
"language": "en",
"url": "https://mathoverflow.net/questions/396250",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Integrality of a sequence formed by sums Consider the following sequence defined as a sum
$$a_n=\sum_{k=0}^{n-1}\frac{3^{3n-3k-1}\,(7k+8)\,(3k+1)!}{2^{2n-2k}\,k!\,(2k+3)!}.$$
QUESTION. For $n\geq1$, is the sequence of rational numbers $a_n$ always integral?
| Here is another proof, inspired by Tewodros Amdeberhan's. We represent the sum as a constant term in a power series.
To represent $(7k+8) \frac{(3k+1)!}{k!\,(2k+3)!}$ as a constant term, we need to express it as a linear combination of binomial coefficients. To do this we express $7k+8$ as a linear combination of $(2k+... | {
"language": "en",
"url": "https://mathoverflow.net/questions/398037",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 4,
"answer_id": 3
} |
How to find the asymptotics of a linear two-dimensional recurrence relation Let $d$ be a positive number.
There is a two dimensional recurrence relation as follow:
$$R(n,m) = R(n-1,m-1) + R(n,m-d)$$
where
$R(0,m) = 1$ and $R(n,0) = R(n,1) = \cdots = R(n, d-1) = 1$ for all $n,m>0$.
How to analyze the asymptotics of $R(n... | This is to complement Blanco's answer by showing that
\begin{equation*}
R(n,kn)=\exp\{(C_{k,d}+o(1))\,n\} \tag{0}\label{0}
\end{equation*}
(as $n\to\infty$), where $k\ge1$ and $d\ge1$ (are fixed),
\begin{equation*}
C_{k,d}:=\frac k{1+y_{k,d}\,d}\,\big(\ln(1+y_{k,d})+y_{k,d}\,\ln(1+1/y_{k,d})\big),
\end{equation... | {
"language": "en",
"url": "https://mathoverflow.net/questions/416269",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 1
} |
Is equation $xy(x+y)=7z^2+1$ solvable in integers? Do there exist integers $x,y,z$ such that
$$
xy(x+y)=7z^2 + 1 ?
$$
The motivation is simple. Together with Aubrey de Grey, we developed a computer program that incorporates all standard methods we know (Hasse principle, quadratic reciprocity, Vieta jumping, search for ... | There is no solution.
It is clear that at least one of $x$ and $y$ is positive and that neither is divisible by 7. We can assume that $a := x > 0$. The equation implies that there are integers $X$, $Y$ such that
$$ X^2 - 7 a Y^2 = a (4 + a^3) $$
(with $X = a (a + 2y)$ and $Y = 2z$).
First consider the case that $a$ is ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/420896",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "31",
"answer_count": 2,
"answer_id": 0
} |
Division problem Are there infinitely many pairs of positive integers $(a,b)$ such that $2(6a+1)$ divides $6b^2+6ab+b-6a^2-2a-3$? That is, if there are infinitely many different $a$ and for which at least one value of $b$ can be found for a given $a$. Some $a$ values are $0,2,3,7,11,17....$
I think that the answer is y... | If $6a+1$ is a prime and a quadratic residue modulo $17$ (which is true for infinitely many values of $a$), then there are infinitely many positive integers $b$ with the required property.
First observe that $b$ is good if and only if $$f(a,b):=6b^2+6ab+b-6a^2-2a-3$$ is even and divisible by $6a+1$. Hence $b$ must be o... | {
"language": "en",
"url": "https://mathoverflow.net/questions/436310",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Parametric Solvable Septics? Known parametric solvable septics are,
$$x^7+7ax^5+14a^2x^3+7a^3x+b=0\tag{1}$$
$$x^7 + 21x^5 + 35x^3 + 7x + a(7x^6 + 35x^4 + 21x^2 + 1)=0\tag{2}$$
$$x^7 - 2x^6 + x^5 - x^4 - 5x^2 - 6x - 4 + n(x - 1)x^2(x + 1)^2=0\tag{3}$$
$$x^7 + 7x^6 - 7\beta x^2 + 28\beta x + 2\beta(n - 13)=0\tag{4}$$
$$... | There is a parametric family of cyclic septics that obey
$$x_1 x_2 + x_2 x_3 + \dots + x_7 x_1 - (x_1 x_3 + x_3 x_5 + \dots + x_6 x_1) = 0\tag1$$
as the Hashimoto-Hoshi septic,
$$\small x^7 - (a^3 + a^2 + 5a + 6)x^6 +
3(3a^3 + 3a^2 + 8a + 4)x^5 + (a^7 + a^6 + 9a^5 - 5a^4 - 15a^3 - 22a^2 -
36a - 8)x^4 - a(a^... | {
"language": "en",
"url": "https://mathoverflow.net/questions/145278",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Go I Know Not Whither and Fetch I Know Not What Next day: apparently my original question is harder, by far, than the other bits. So: it is a finite check, I was able to confirm by computer that, if the polynomial below satisfies $$ f(a,b,c,d) \equiv 0 \pmod {27}, \;\; \mbox{THEN} \; \; a,b,c,d \equiv 0 \pmod 3, $$ an... | Yes, this is a field norm; it is the norm of $a + b \sqrt{3} + c \sqrt{5} + d \sqrt{15}$, from $K = \mathbb{Q}(\sqrt{3}, \sqrt{5})$ down to $\mathbb{Q}$. Note that $a+b \sqrt{3} + c \sqrt{5} + d \sqrt{15}$ acts on the basis $(1, \sqrt{3}, \sqrt{5}, \sqrt{15})$ by
$$a \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/180987",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
} |
Find all solution $a,b,c$ with $(1-a^2)(1-b^2)(1-c^2)=8abc$ Two years ago, I made a conjecture on stackexchange:
Today, I tried to find all solutions in integers $a,b,c$ to
$$(1-a^2)(1-b^2)(1-c^2)=8abc,\quad a,b,c\in \mathbb{Q}^{+}.$$
I have found some solutions, such as
$$(a,b,c)=(5/17,1/11,8/9),(1/7,5/16,9/11),(3/4... | The original proposer asks for "simple methods". Simplicity, like beauty, is in the eye of the beholder. I am sure
that Noam Elkies and Joe Silverman feel their answers are extremely simple. The following discussion is, in my humble opinion,
simpler.
We can express the underlying equation as a quadratic in $a$,
\begin{... | {
"language": "en",
"url": "https://mathoverflow.net/questions/208485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 4,
"answer_id": 3
} |
sum, integral of certain functions While working on some research, I have encountered an infinite series and its improper integral analogue:
\begin{align}\sum_{m=1}^{\infty}\frac1{\sqrt{m(m+1)(m+2)+\sqrt{m^3(m+2)^3}}}&=\frac12+\frac1{\sqrt{2}}, \\
\int_0^{\infty}\frac{dx}{\sqrt{x(x+1)(x+2)+\sqrt{x^3(x+2)^3}}}&=2.\end{a... | For the integral, notice that the expression under the square root is
$$ x(x+1)(x+2)+x(x+2)\sqrt{x(x+2)} = \frac12\,x(x+2)(\sqrt x+\sqrt{x+2})^2. $$
Consequently,
\begin{align*}
\frac1{\sqrt{x(x+1)(x+2)+x(x+2)\sqrt{x(x+2)}}}
&= \frac{\sqrt 2}{(\sqrt x+\sqrt{x+2}) \sqrt{x(x+2)}} \\
&= \frac1{\sqrt 2}... | {
"language": "en",
"url": "https://mathoverflow.net/questions/257982",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Question on a generalisation of a theorem by Euler We call an integer $k\geq 1$ good if for all $q\in\mathbb{Q}$ there are $a_1,\ldots, a_k\in \mathbb{Q}$ such that $$q = \prod_{i=1}^k a_i \cdot\big(\sum_{i=1}^k a_i\big).$$
Euler showed that $k=3$ is good.
Is the set of good positive integers infinite?
| I suspect that $k = 4$ is good, but am not sure how to prove it. However, every positive integer $k \geq 5$ is good. This follows from the fact (see the proof of Theorem 1 from this preprint) that for any rational number $x$, there are rational numbers $a$, $b$, $c$, $d$ so that $a+b+c+d = 0$ and $abcd = x$. In particu... | {
"language": "en",
"url": "https://mathoverflow.net/questions/302933",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 2,
"answer_id": 0
} |
How to find the analytical representation of eigenvalues of the matrix $G$? I have the following matrix arising when I tried to discretize the Green function， now to show the convergence of my algorithm I need to find the eigenvalues of the matrix $G$ and show it has absolute value less than 1 for certain choices of $N... | It's straightforward to show that this is the inverse of $1/(N+1)$ times the tridiagonal matrix $T_N$ with $-2$ on its main diagonal and $1$ on its super- and sub-diagonals.
Let $t_N$ be the characteristic polynomial of $T_N$. We have $t_0(x)=1$, $t_1(x)=x+2$, and by cofactor expansion $t_N(x)=(x+2)t_{N-1}(x)-t_{N-2}(x... | {
"language": "en",
"url": "https://mathoverflow.net/questions/308835",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Matrix rescaling increases lowest eigenvalue? Consider the set $\mathbf{N}:=\left\{1,2,....,N \right\}$ and let $$\mathbf M:=\left\{ M_i; M_i \subset \mathbf N \text{ such that } \left\lvert M_i \right\rvert=2 \text{ or }\left\lvert M_i \right\rvert=1 \right\}$$
be the set of all subsets of $\mathbf{N}$ that are of car... |
Claim. $\lambda_\min(A_N) \le 4\lambda_\min(B_N)$.
Proof.
Let $C_N:=\bigl[\tfrac{1}{|M_i||M_j|}\bigr]$. Then, $B_N = A_N \circ C_N$, where $\circ$ denotes the Hadamard product. Observe that by construction both $A_N$ and $C_N$ are positive semidefinite, so $B_N$ is also psd. Let's drop the subscript $N$ for brevity. ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/313470",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
} |
Asymptotic Expansion of Bessel Function Integral I have an integral:
$$I(y) = \int_0^\infty \frac{xJ_1(yx)^2}{\sinh(x)^2}\ dx $$
and would like to asymptotically expand it as a series in $1/y$. Does anyone know how to do this? By numerically computing the integral it appears that
$$I(y) = \frac 12 - \frac 1 {\pi y}+ \f... | Inserting the Mellin-Barnes representation for the square of the Bessel function (DLMF),
\begin{equation}
J_{1}^2\left(xy\right)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i
\infty}\frac{\Gamma\left(-t\right)\Gamma\left(2t+3\right)}{\Gamma^2\left(t+2\right)\Gamma%
\left(t+3\right)}\left(\frac{xy}{2}\right)^{2t+2}\,dt
\end{eq... | {
"language": "en",
"url": "https://mathoverflow.net/questions/315264",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
eigenvalues of a symmetric matrix I have a special $N\times N$ matrix with the following form. It is symmetric and zero row (and column) sums.
$$K=\begin{bmatrix}
k_{11} & -1 & \frac{-1}{2} & \frac{-1}{3} & \frac{-1}{4} & \ldots &
\frac{-1}{N-2} & \frac{-1}{N-1} & \\
-1 & k_{22} & \frac{-1}{2} & \frac{-1}{3} & \frac{-1... | Phillip Lampe seems to be correct. Here are the eigenvalues and eigenvectors computed by hand:
Let $k_1 = 2 + \tfrac12 + \cdots + \tfrac{1}{N-1}$, then:
$\lambda_0 = 0$ with eigenvector all ones (by construction).
$\lambda_1 = k_{1}$ with eigenvector $\begin{bmatrix}-1& 1& 0&\cdots& 0\end{bmatrix}^T$
$\lambda_2 = k_1-... | {
"language": "en",
"url": "https://mathoverflow.net/questions/324165",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Representation of $4\times4$ matrices in the form of $\sum B_i\otimes C_i$ Every matrix $A\in M_4(\mathbb{R})$
can be represented in the form of $$A=\sum_{i=1}^{n(A)} B_i\otimes C_i$$ for $B_i,C_i\in M_2(\mathbb{R})$.
What is the least uniform upper bound $M$ for such $n(A)$? In other words, what is the least integer $... | Because $M_4(\mathbb R) = M_2(\mathbb R) \otimes M_2(\mathbb R)$ as vector spaces (and as algebras, but we won't use this), we can replace $M_2(\mathbb R)$ by an arbitrary $4$-dimensional vector space $V$ and $M_4(\mathbb R)$ by $V \otimes V$. We can represent elements of $V\otimes V$ conveniently as $4\times 4$ matric... | {
"language": "en",
"url": "https://mathoverflow.net/questions/331525",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Primality test for specific class of $N=8kp^n-1$ My following question is related to my question here
Can you provide a proof or a counterexample for 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 $N=8kp^n-1$ such that $k>0$ , $3 \not... | This is a partial answer.
This answer proves that if $N$ is a prime, then $S_{n-2}\equiv 0\pmod N$.
Proof :
It can be proven by induction that
$$S_i=(2-\sqrt 3)^{2kp^{i+2}}+(2+\sqrt 3)^{2kp^{i+2}}\tag1$$
Using $(1)$ and $2\pm\sqrt 3=\bigg(\frac{\sqrt{6}\pm\sqrt 2}{2}\bigg)^2$, we get
$$\begin{align}&2^{N+1}S_{n-2}^2-2... | {
"language": "en",
"url": "https://mathoverflow.net/questions/361489",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How small the radical of $xy(x+y)uv(u+v)$ can be infinitely often? Let $x,y,u,v$ be positive integers with $x,y$ coprime and $u,v$ coprime
( $xy,uv$ not necessarily coprime). Assume $x+y \ne u+v$.
How small the radical of $xy(x+y)uv(u+v)$ can be infinitely often?
Can we get $O(|(x+y)(u+v)|^{1-C})$ for $C>0$?
These are ... | Here is a solution where the radical is $O(k^9)$ and $(x+y)(u+v)=O(k^{12})$
The idea is that $x,y,z=a^2,b^2,c^2$ for a Pythagorean triple and $u,v,u+v=A^2,B^2,C^2$ for another with $C=c^2.$ I used the most familiar type of triple (hypotenuse and long leg differ by $1$), there might be others that do better, or special ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/377124",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Is there a nonzero solution to this infinite system of congruences? Is there a triple of nonzero even integers $(a,b,c)$ that satisfies the following infinite system of congruences?
$$
a+b+c\equiv 0 \pmod{4} \\
a+3b+3c\equiv 0 \pmod{8} \\
3a+5b+9c\equiv 0 \pmod{16} \\
9a+15b+19c\equiv 0 \pmod{32} \\
\vdots \\
s_na + t_... | $u_n=s_na + t_nb + s_{n+1}c$ satisfies the same recurrence relation as $s_n$ and $t_n$: $u_n = u_{n-1} +2u_{n-2} + 4u_{n-3}$. The question is whether $2^{n+1}\mid u_n$.
Since $v_n=u_n/2^{n+1}$ satisfies
$v_n = \displaystyle\frac{v_{n-1} +v_{n-2} + v_{n-3}}{2}$
the answer is affirmative only if there are $v_0, v_1, v_2$... | {
"language": "en",
"url": "https://mathoverflow.net/questions/381057",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
An explicit equation for $X_1(13)$ and a computation using MAGMA By a general theory $X_1(13)$ is smooth over $\mathbb{Z}[1/13]$, and so is its Jacobian $J$.
And the hyperelliptic curve given by an affine model $y^2 = x^6 - 2x^5 + x^4 -2x^3 + 6x^2 -4x + 1$ is $X_1(13)$.
However, according to MAGMA, $J$ is bad at $2$.
... | To get a model with good reduction at $2$, take $y = 2Y + x^3 + x^2 + 1$,
subtract $(x^3+x^2+1)^2$ from both sides, and divide by $4$ to get
$$ Y^2 + (x^3+x^2+1) \, Y = -x^5-x^3+x^2-x. $$
(A similar tactic of un-completing the square
is well-known for elliptic curves.)
| {
"language": "en",
"url": "https://mathoverflow.net/questions/385038",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Improper integral $\int_0^\infty\frac{x^{2n+1}}{(1+x^2)^2(2+x^2)^N}dx,\ \ \ n\le N$ How can I evaluate this integral?
$$\int_0^\infty\frac{x^{2n+1}}{(1+x^2)^2(2+x^2)^N}dx,\ \ \ n\le N$$
Maybe there is a recurrence relation for the integral?
| Let us renumber $N=n+L$ and let $K_{n,L} = I_{n,n+L} = \frac{1}{2} \int_0^\infty \frac{y^n}{(1+y)^2 (2+y)^{n+L}} \, dy$, which is the desired integral after the variable change $y=x^2$. Let $K(s,t) = \sum_{L=0}^\infty \sum_{n=0}^\infty K_{n,L} s^L t^n$. For small enough $s$ and $t$, the integrands converge uniformly ov... | {
"language": "en",
"url": "https://mathoverflow.net/questions/393753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 2
} |
General formulas for derivative of $f_n(x)=\dfrac{ax^n+bx^{n-1}+cx^{n-2}+\cdots}{a'x^n+b'x^{n-1}+c'x^{n-2}+\cdots},\quad a'\neq0$ For the function $f_1(x)=\dfrac{ax+b}{a'x+b'},\quad a'\neq0$ , we have $$f_1'(x)=\dfrac{\begin{vmatrix}{a} && {b} \\ {a'} && {b'}\end{vmatrix}}{(a'x+b')^2}$$
For $f_2(x)=\dfrac{ax^2+bx+c}{a'... | If $f_n=\frac{P_n(x)}{Q_n(x)}$ and $P_n=\sum a_kx^k$,
$Q_n=\sum b_kx^k$, then
$$f'_n=\frac{\begin{vmatrix}{P'} && {Q'} \\ {P} && {Q}\end{vmatrix} }{Q^2}$$
Breaking the determinant on the numerator gives $$\sum_{j\geq 0} \left(\sum_{k+r=j+1} k\begin{vmatrix}{a_{k}} && {b_k} \\ {a_{j+1-k}} && {b_{j+1-k}}\end{vmatrix} \ri... | {
"language": "en",
"url": "https://mathoverflow.net/questions/396250",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Any hints on how to prove that the function $\lvert\alpha\;\sin(A)+\sin(A+B)\rvert - \lvert\sin(B)\rvert$ is negative over the half of the total area? I have this inequality with $0<A,B<\pi$ and a real $\lvert\alpha\rvert<1$:
$$ f(A,B):=\bigl|\alpha\;\sin(A)+\sin(A+B)\bigr| - \bigl| \sin(B)\bigr| < 0$$
Numerically, I... | Let us assume $\alpha\in[0,1)$ (the case of $\alpha\in(-1,0]$ is similar). As $\sin B>0$ for $B\in (0,\pi)$, the inequality $f(A,B)<0$ amounts to
$$
\alpha\sin A<\sin B-\sin(A+B),\quad -[\sin B+\sin(A+B)]<\alpha\sin A.\quad (\star)
$$
Notice that
$\sin A=2\sin\left(\frac{A}{2}\right)\cos\left(\frac{A}{2}\right)$,
$\si... | {
"language": "en",
"url": "https://mathoverflow.net/questions/401878",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 0
} |
Can we cover the unit square by these rectangles? The following question was a research exercise (i.e. an open problem) in R. Graham, D.E. Knuth, and O. Patashnik, "Concrete Mathematics", 1988, chapter 1.
It is easy to show that
$$\sum_{1 \leq k } \left(\frac{1}{k} \times \frac{1}{k+1}\right) = \sum_{1 \leq k } \left(... | it is hard to prove this problem directly, but it is not hard to prove (as someone mentioned in some comments) :
If (n-1) rectangles has been put into the 1x1 square, then all rectangles can be put into the square of length (1+1/n)
Proof:
we have put (n-1) rectangles into unit square
denote $P_n = \frac{1}{n} \times \f... | {
"language": "en",
"url": "https://mathoverflow.net/questions/34145",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "95",
"answer_count": 6,
"answer_id": 5
} |
General integer solution for $x^2+y^2-z^2=\pm 1$ How to find general solution (in terms of parameters) for diophantine equations
$x^2+y^2-z^2=1$ and $x^2+y^2-z^2=-1$?
It's easy to find such solutions for $x^2+y^2-z^2=0$ or $x^2+y^2-z^2-w^2=0$ or $x^2+y^2+z^2-w^2=0$, but for these ones I cannot find anything relevant.
| I believe the general solution to $x^2+y^2-z^2=1$ is $x=(rs+tu)/2$, $y=(rt-su)/2$, $z=(rs-tu)/2$, where $rt+su=2$.
EDIT: Solutions to $x^2+y^2+1=z^2$ can be obtained by choosing $a$, $b$, $c$, $d$ such that $ad-bc=1$ and then letting $x=(a^2+b^2-c^2-d^2)/2$, $y=ac+bd$, $z=(a^2+b^2+c^2+d^2)/2$, though I'm not sure you ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/65957",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 1
} |
Fermat's proof for $x^3-y^2=2$ Fermat proved that $x^3-y^2=2$ has only one solution $(x,y)=(3,5)$.
After some search, I only found proofs using factorization over the ring $Z[\sqrt{-2}]$.
My question is:
Is this Fermat's original proof? If not, where can I find it?
Thank you for viewing.
Note: I am not expecting to fin... | Here is how Fermat probably did it (it is how I did it - not all of the steps were needed but I have to believe this was close to Fermat's thought process).
Any prime of the form $8n+1$ or $8n+3$ can be written in the form $a^2 +2b^2$. This is proved with descent techniques once realizes that $-2$ and $1$ are squares ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/142220",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "30",
"answer_count": 5,
"answer_id": 3
} |
The relationship between the dilogarithm and the golden ratio Among the values for which the dilogarithm and its argument can both be given in closed form are the following four equations:
$Li_2( \frac{3 - \sqrt{5}}{2}) = \frac{\pi^2}{15} - log^2( \frac{1 +\sqrt{5}}{2} )$ (1)
$Li_2( \frac{-1 + \sqrt{5}}{2}) = \frac{\pi... | According to Maple
$$ {\rm polylog}\left(2, \dfrac{1+\sqrt{5}}{2} \right) = \dfrac{7 \pi^2}{30} - \dfrac{1}{2} \log^2 \left(\dfrac{1+\sqrt{5}}{2}\right) - \log \left(\dfrac{1+\sqrt{5}}{2}\right) \log \left(\dfrac{1-\sqrt{5}}{2}\right)$$
Of course $\log((1-\sqrt{5})/2) = \log(-1/\phi) = - \log (\phi) + i \pi$
| {
"language": "en",
"url": "https://mathoverflow.net/questions/144322",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
} |
Number of Permutations? Edit: This is a modest rephrasing of the question as originally stated below the fold: for $n \geq 3$, let $\sigma \in S_n$ be a fixed-point-free permutation. How many fixed-point-free permutations $\tau$ are there such that $\sigma \tau^{-1}$ is also fixed-point free? As the original post shows... | A formula for the answer to this question is given in formula (3) of J. Riordan, Three-line Latin rectangles, Amer. Math. Monthly 51, (1944), 450–452. Riordan doesn't really include a proof, though it's not too hard to see how the formula follows from the theory of rook polynomials. (He refers to a paper of Kaplansky,... | {
"language": "en",
"url": "https://mathoverflow.net/questions/144899",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 5,
"answer_id": 2
} |
Binomial Identity I recently noted that
$$\sum_{k=0}^{n/2} \left(-\frac{1}{3}\right)^k\binom{n+k}{k}\binom{2n+1-k}{n+1+k}=3^n$$
Is this a known binomial identity? Any proof or reference?
| I rewrote your identity in an equivalent form
$$
\sum_{k=0}^{n/2} (-1)^k 3^{n-k} \binom{n+k}{n, k} \binom{2n-k+1}{n-2k, n+k+1} = 3^{2n} ,
$$
and attempted to construct a proof by induction on $n$. I did not succeed, but discovered an interesting hurdle: to
prove the equally curious identity
$$
\sum_{k=0}^{n/2} (-1)^... | {
"language": "en",
"url": "https://mathoverflow.net/questions/155402",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 1
} |
What is the value of the infinite product: $(1+ \frac{1}{1^1}) (1+ \frac{1}{2^2}) (1+ \frac{1}{3^3}) \cdots $? What is the value of the following infinite product?
$$\left(1+ \frac{1}{1^1}\right) \left(1+ \frac{1}{2^2}\right) \left(1+ \frac{1}{3^3}\right) \cdots $$
Is the value known?
| I'm not sure what the criterium for a full answer is here, so here a technique for $(1+c_k)$ kind of products, turning the infinite product into an infinite sum:
Via telescoping, for friendly $a_n$ and any $m$, we have
$$\lim_{n\to\infty}a_n=a_m+\sum_{n=m}^\infty\left(\dfrac{a_{n+1}}{a_n}-1\right)\,{a_n}.$$
So define
... | {
"language": "en",
"url": "https://mathoverflow.net/questions/200815",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Create matrix containing values in [0,1] where sum of all diagonals and anti-diagonals is fixed The problem I am facing sounds at first glance pretty simple. However, as very often, it seems more complicated than I first assumed:
I want to calculate a matrix $P = (p_{j,k}) \in \mathbb{R}^{n \times n}$, $n\in\mathbb{N}$... | The given constraints are a system of linear inequalities, so you can find a feasible solution (or prove that none exists) by feeding these constraints to a linear program (LP) solver with some arbitrarily chosen objective function.
| {
"language": "en",
"url": "https://mathoverflow.net/questions/203588",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Are there some known identities of elliptic polylogarithms similar to the Abel identity of polylogarithm? Let
\begin{align}
Li_2(z) = \sum_{n=1}^{\infty} \frac{z^n}{n^2}.
\end{align}
This polylogarithm satisfies the following Abel identity:
\begin{align}
& Li_2(-x) + \log x \log y \\
& + Li_2(-y) + \log ( \frac{1+y}... | The 5-term relation is a special case of the Rogers identity: theorem 8.14 here. This is a degenerate version of the Bloch relation for elliptic dilogarithm (see page 30 here).
| {
"language": "en",
"url": "https://mathoverflow.net/questions/261765",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
What is known about the plethysm $\text{Sym}^d(\bigwedge^3 \mathbb{C}^6)$ What is known about the plethysm $\text{Sym}^d(\bigwedge^3 \mathbb{C}^6)$ as a representation of $\text{GL}(6)$? It is my understanding that this should be multiplicity-free. I tried computing it using the Schur Rings package in Macaulay2 and I c... | No, it is not multiplicity-free. Already for $d=6$, this representation contains the Schur functor $S^{4,4,4,2,2,2}$ twice. This can be easily checked in Magma (even the online calculator) issuing the commands
Q := Rationals();
s := SFASchur(Q);
s.[6]~s.[1,1,1];
Ignoring the Schur functors that vanish on $\mathbb{C}^6... | {
"language": "en",
"url": "https://mathoverflow.net/questions/294184",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 0
} |
Is there a sense in which one could expand $\frac{\sin (x/\epsilon )}{x} $ in powers of $\epsilon $? A standard representation of the $\delta $-distribution is
$$
\pi \delta (x) = \lim_{\epsilon \searrow 0} \frac{\sin (x/\epsilon )}{x}
$$
Is there a sense in which this could be seen as the leading term in an expansion ... | The expansion around point $a$ in positive powers exists, but what is the general form of the term and whether it is convergent requires further research.
$$\frac{\sin (x/\epsilon)}x=\frac{\sin \left(\frac{x}{a}\right)}{x}-\frac{(\epsilon -a) \cos \left(\frac{x}{a}\right)}{a^2}+\frac{(\epsilon -a)^2 \left(2 a \cos \lef... | {
"language": "en",
"url": "https://mathoverflow.net/questions/350697",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
How to prove the determinant of a Hilbert-like matrix with parameter is non-zero Consider some positive non-integer $\beta$ and a non-negative integer $p$. Does anyone have any idea how to show that the determinant of the following matrix is non-zero?
$$
\begin{pmatrix}
\frac{1}{\beta + 1} & \frac{1}{2} & \frac{1}{3} &... | I think the reference "Advanced Determinant Calculus" has a pointer to the answer. But I'll still elaborate for it is ingenious.
Suppose $x_i$'s and $y_j$'s, $1\leq i,j \leq N$, are numbers such that $x_i+y_j\neq 0$ for any $i,j$ combination, then the following identity (called Cauchy Alternant Identity) holds good:
$... | {
"language": "en",
"url": "https://mathoverflow.net/questions/358175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 2,
"answer_id": 0
} |
Improper integral $\int_0^\infty\frac{x^{2n+1}}{(1+x^2)^2(2+x^2)^N}dx,\ \ \ n\le N$ How can I evaluate this integral?
$$\int_0^\infty\frac{x^{2n+1}}{(1+x^2)^2(2+x^2)^N}dx,\ \ \ n\le N$$
Maybe there is a recurrence relation for the integral?
| Recurrence (3) in my other answer on this page also follows immediately from the same recurrence for the respective integrands! :-)
Concerning the case $n=0$: the recurrence for the $M_k$'s follows immediately from the definition of $M_k$ and the trivial identity $y^{-k}-y^{-k-1}=y^{-k-1}(y-1)$.
To make this answer in... | {
"language": "en",
"url": "https://mathoverflow.net/questions/393753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 3
} |
Any hints on how to prove that the function $\lvert\alpha\;\sin(A)+\sin(A+B)\rvert - \lvert\sin(B)\rvert$ is negative over the half of the total area? I have this inequality with $0<A,B<\pi$ and a real $\lvert\alpha\rvert<1$:
$$ f(A,B):=\bigl|\alpha\;\sin(A)+\sin(A+B)\bigr| - \bigl| \sin(B)\bigr| < 0$$
Numerically, I... | This is equivalent to
\begin{align}
|\alpha \sin A + \sin(A+B)|&<|\sin B|\\
((\alpha+\cos B) \sin A + \cos A \sin B)^2&<(\sin B)^2\\
((\alpha + \cos B)^2-\sin^2 B)\sin^2 A
&<-2(\alpha+\cos\beta)\sin A \cos A \sin B\\
\frac{(\alpha + \cos B)^2-\sin^2 B}{\sin B}
&<\frac{-2(\alpha+\cos\beta)\cos A}{\sin A}\\
\end{align}
S... | {
"language": "en",
"url": "https://mathoverflow.net/questions/401878",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 1
} |
Inverse Mellin transform of 3 gamma functions product I want to calculate the inverse Mellin transform of products of 3 gamma functions.
$$F\left ( s \right )=\frac{1}{2i\pi}\int \Gamma(s)\Gamma (2s+a)\Gamma( 2s+b)x^{-s}ds$$
Above contour integral has 3 poles.
$$s_{1}=-n$$
$$s_{2}=-\left ( \frac{n+a}{2} \right )$$
$$... | To avoid all poles in the Mellin inversion formula you want to integrate along the line $\int_{\gamma-i\infty}^{\gamma+i\infty}ds$ where $\gamma>\max(0,-a/2,-b/2)$; then Mathematica gives the result in terms of the Meijer G-function:
\begin{align}
F( x )=&\frac{1}{2i\pi}\int_{\gamma-i\infty}^{\gamma+i\infty} \Gamma(s)\... | {
"language": "en",
"url": "https://mathoverflow.net/questions/420794",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Ramanujan and algebraic number theory One out of the almost endless supply of identities discovered by Ramanujan
is the following:
$$ \sqrt[3]{\rule{0pt}{2ex}\sqrt[3]{2}-1} = \sqrt[3]{\frac19} - \sqrt[3]{\frac29} + \sqrt[3]{\frac49}, $$
which has the following interpretation in algebraic number theory: the fundamental ... | $$(7 \sqrt[3]{20} - 19)^{1/6} = \ \sqrt[3]{\frac{5}{3}}
- \sqrt[3]{\frac{2}{3}},$$
$$\left( \frac{3 + 2 \sqrt[4]{5}}{3 - 2 \sqrt[4]{5}}
\right)^{1/4}= \ \ \frac{\sqrt[4]{5} + 1}{\sqrt[4]{5} - 1},$$
$$\left(\sqrt[5]{\frac{1}{5}} + \sqrt[5]{\frac{4}{5}}\right)^{1/2}
= \ \ (1 + \sqrt[5]{2} + \sqrt[5]{8})^{1/5} = \ \ ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/43388",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "30",
"answer_count": 1,
"answer_id": 0
} |
A lower bound of a particular convex function Hello,
I suspect this reduces to a homework problem, but I've been a bit hung up on it for the last few hours. I'm trying to minimize the (convex) function $f(x) = 1/x + ax + bx^2$ , where $x,a,b>0$. Specifically, I'm interested in the minimal objective function value as ... | The first inequality is true. Write
$$f=\frac{a}{a+b}f_0+\frac{b}{a+b}f_1,$$
where $f_0$ and $f_1$ correspond to the case $b=0$ and $a=0$, respectively. You know that $f_0\ge2\sqrt a$ and $f_1\ge\frac{3\cdot2^{1/3}}{2}b^{1/3}$. This implies
$$f\ge\frac{a}{a+b}2\sqrt a+\frac{b}{a+b}\frac{3\cdot2^{1/3}}{2}b^{1/3}.$$
| {
"language": "en",
"url": "https://mathoverflow.net/questions/61946",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
General integer solution for $x^2+y^2-z^2=\pm 1$ How to find general solution (in terms of parameters) for diophantine equations
$x^2+y^2-z^2=1$ and $x^2+y^2-z^2=-1$?
It's easy to find such solutions for $x^2+y^2-z^2=0$ or $x^2+y^2-z^2-w^2=0$ or $x^2+y^2+z^2-w^2=0$, but for these ones I cannot find anything relevant.
| I think that the solutions to $x^2+y^2-z^2=-1$ are $x=RT-SU,y=RU+ST$ where $R^2+S^2-T^2-U^2=2$ then $z=R^2+S^2-1=T^2+U^2+1$ On the surface this looks similar to the solutions to the $+1$ case. However these are quite a bit rarer and depend on the locations of the primes.
As we know, an integer can be uniquely written... | {
"language": "en",
"url": "https://mathoverflow.net/questions/65957",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 2
} |
Computing the centers of Apollonian circle packings The radii of an Apollonian circle packing are computed from the initial curvatures e.g. (-10, 18, 23, 27) solving Descartes equation $2(a^2+b^2+c^2+d^2)=(a+b+c+d)^2$ and using the four matrices to generate more solutions
$$
\left[\begin{array}{cccc} -1 & 2 & 2 & 2 \\... | $$2(a^2+b^2+c^2+d^2)=(a+b+c+d)^2$$
$$a=4k(k+s)$$
$$b=4s(k+s)$$
$$c=p^2+k^2+s^2+2pk+2ps-2ks$$
$$d=p^2+k^2+s^2-2pk-2ps-2ks$$
| {
"language": "en",
"url": "https://mathoverflow.net/questions/88353",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 5,
"answer_id": 4
} |
The relationship between the dilogarithm and the golden ratio Among the values for which the dilogarithm and its argument can both be given in closed form are the following four equations:
$Li_2( \frac{3 - \sqrt{5}}{2}) = \frac{\pi^2}{15} - log^2( \frac{1 +\sqrt{5}}{2} )$ (1)
$Li_2( \frac{-1 + \sqrt{5}}{2}) = \frac{\pi... | Wikipedia says $$Li_2\left({1+\sqrt5\over2}\right)={\pi^2\over10}-\log^2{\sqrt5-1\over2}$$
| {
"language": "en",
"url": "https://mathoverflow.net/questions/144322",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
} |
$\zeta(0)$ and the cotangent function In preparing some practice problems for my complex analysis students, I stumbled across the following. It is not hard to show, using Liouville's theorem, that
$$\pi\cot(\pi z)=\frac{1}{z}+\sum_{n=1}^\infty\left(\frac{1}{z+n}+\frac{1}{z-n}\right),$$
which implies that
$$-\frac{\pi z... | Here is an explanation based on the Euler-Maclaurin summation formula.
(Or rather, since we'll only ever need two terms of the Euler-Maclaurin summation, it's really more or less just "the trapezoid rule".)
I think it's a good explanation because it sticks to the general structure of the argument outlined in the questi... | {
"language": "en",
"url": "https://mathoverflow.net/questions/188371",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "37",
"answer_count": 2,
"answer_id": 0
} |
Determinant of a matrix filled with elements of the Thue–Morse sequence Let $n$ be a positive integer. Suppose we fill a square matrix $n\times n$ row-by-row with the first $n^2$ elements of the Thue–Morse sequence (with indexes from $0$ to $n^2-1$). Let $\mathcal D_n$ be the determinant of this matrix. For example,
$$... | not an answer just the result of a computation. The following plot shows for each $n$ the minimal number $k$ such that the first $k$ rows are linearly dependent. The question is to find all $n$ such that $k=n+1$.
| {
"language": "en",
"url": "https://mathoverflow.net/questions/314742",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 1,
"answer_id": 0
} |
How to prove the determinant of a Hilbert-like matrix with parameter is non-zero Consider some positive non-integer $\beta$ and a non-negative integer $p$. Does anyone have any idea how to show that the determinant of the following matrix is non-zero?
$$
\begin{pmatrix}
\frac{1}{\beta + 1} & \frac{1}{2} & \frac{1}{3} &... | Rows linearly dependent means for some $c_1$, $\ldots$, $c_{p+1}$ the non-zero rational function
$\sum_{k=1}^{p+1} \frac{c_k}{x+k}$ has $p+1$ roots $\beta$, $1$, $2$, $\ldots$, $p$, not possible, since its numerator has degree at most $p$.
| {
"language": "en",
"url": "https://mathoverflow.net/questions/358175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 2,
"answer_id": 1
} |
Norms in quadratic fields This should be well-known, but I can't find a reference (or a proof, or a counter-example...). Let $d$ be a positive square-free integer. Suppose that there is no element in the ring of integers of $\mathbb{Q}(\sqrt{d})$ with norm $-1$. Then I believe that no element of $\mathbb{Q}(\sqrt{d})$ ... | Dirichlet's version of Gauss composition is in the book by Cox, (page 49 in first) with a small typo corrected in the second edition.
For our purpose, duplication, it has a better look to equate $a=a'$ from the start, with $\gcd(a,b) = 1$ sufficing,
$$ \left( ax^2 +bxy+ acy^2 \right) \left( aw^2 +bwz+ acz^2 \right) =... | {
"language": "en",
"url": "https://mathoverflow.net/questions/369846",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 3,
"answer_id": 0
} |
How the solve the equation $\frac{(a+b\ln(x))^2}{x}=c$ I need to solve the equation
$$\frac{(a+b\ln(x))^2}{x}=c$$
where $a$, $b$, and $c$ are given. It is known that $a$ and $b$ are fixed and satisfy some condition such that the left hand side is decreasing. So $x$ is uniquely determined by $c$ when $c$ is chosen in ce... | Step-by-step solution with Lambert W. The goal is to get something
of the form $\color{red}{ue^u = v}$ then re-write it as $\color{blue}{u=W(v)}$.
$$
\frac{(a+b\ln(x))^2}{x}=c
\\
\frac{(a+b\ln(x))}{\sqrt{x}}=\pm\sqrt{c}
\\
(a+b\ln(x))e^{-\ln(x)/2}=\pm\sqrt{c}
\\
(a+b\ln(x))\exp\left(-\frac{a}{2b}-\frac{\ln(x)}{2}\righ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/410729",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"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.