Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
Infinite series for $1/\pi$. Is it known? Indirect method (associated with a certain problem of electrostatics) indicates that $$\sum\limits_{j=1}^\infty \frac{(2j-3)!!\,(2j-1)!!}{(2j-2)!!\,(2j+2)!!}=\frac{2}{3\pi}.$$ Is this result known?
| Using the standard power series for the complete elliptic integral of the second kind $$E(k) = \frac{\pi}{2} \sum_{j=0}^\infty \left(\frac{(2j)!}{2^{2j}(j!)^2}\right)^2 \frac{k^{2j}}{1-2j},$$
we find
\begin{align*}
\sum\limits_{j=1}^\infty \frac{(2j-3)!!\,(2j-1)!!}{(2j-2)!!\,(2j+2)!!} k^{2j}&=\sum_{j=1}^\infty\frac{-j}... | {
"language": "en",
"url": "https://mathoverflow.net/questions/418193",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
Some Log integrals related to Gamma value Two years ago I evaluated some integrals related to $\Gamma(1/4)$.
First example:
$$(1)\hspace{.2cm}\int_{0}^{1}\frac{\sqrt{x}\log{(1+\sqrt{1+x})}}{\sqrt{1-x^2}} dx=\pi-\frac{\sqrt {2}\pi^{5/2}+4\sqrt{2}\pi^{3/2}}{2\Gamma{(1/4)^{2}}}.$$
The proof I have is based on the followin... | I also played around with this integral. My solution is a bit shorter than the OPs: First use the trick by @Claude and define
$$\tag{1}
I(a)=\int_0^1 \mathrm dx \sqrt\frac{x}{1-x^2}\log(a+\sqrt{1+x}),
$$
such that
$$\tag{2}
I(1) = I(0) + \int_0^1 \mathrm da \, I'(a).
$$
Partial fraction decomposition of $I'(a)$ gives
\... | {
"language": "en",
"url": "https://mathoverflow.net/questions/437979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Euclidean volume of the unit ball of matrices under the matrix norm The matrix norm for an $n$-by-$n$ matrix $A$ is defined as $$|A| := \max_{|x|=1} |Ax|$$ where the vector norm is the usual Euclidean one. This is also called the induced (matrix) norm, the operator norm, or the spectral norm. The unit ball of matrices ... | I worked out the answer for the 2 by 2 case as well.
First, when dealing with 2 by 2 matrices in general, a convenient variable change is:
$$a\rightarrow\frac{w+x}{\sqrt{2}},d\rightarrow\frac{w-x}{\sqrt{2}},c\rightarrow\frac{y-z}{\sqrt{2}},b\rightarrow\frac{y+z}{\sqrt{2}}.$$
Then $a^2+b^2+c^2+d^2 = w^2+x^2+y^2+z^2$. A... | {
"language": "en",
"url": "https://mathoverflow.net/questions/1464",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
"answer_count": 8,
"answer_id": 5
} |
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 ... | As Nishant Chandgotia sugessted: simply write
$f(x) = \left(p\cdot \frac{1}{x} + ax\right) + \left((1-p)\frac{1}{x}+bx^2 \right)$
for some parametr $p\in[0,1]$.
For the first term, minimizer is equal to $ p^{\frac{1}{2}}a^{-\frac{1}{2}}$ and the minimal value is $p^{\frac{1}{2}} 2a^{\frac{1}{2}}$.
For the second therm,... | {
"language": "en",
"url": "https://mathoverflow.net/questions/61946",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
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.
| Let´s take any $x>3$ and choose a,b such that $a<b$,$a-b$ even and $ab=x^2-1$.
Then with $y=(b-a)/2$, $z=(b+a)/2$ we have $x^2 + y^2 = z^2 + 1$.
Particular cases∶
if $x$ even then $x^2 + ((x^2-2)/2)^2 = (x^2/2)^2 + 1$;
if $x$ odd then $x^2 + ((x^2-5)/4)^2 = ((x^2+3)/4)^2 + 1$.
| {
"language": "en",
"url": "https://mathoverflow.net/questions/65957",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 3
} |
Source and context of $\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 dx/(1+x^2)$? Possibly the most striking proof of Archimedes's inequality $\pi < 22/7$ is an integral formula for the difference:
$$
\frac{22}{7} - \pi = \int_0^1 (x-x^2)^4 \frac{dx}{1+x^2},
$$
where the integrand is manifestly positive. This formula is "we... | This integral has a series counterpart
$$\sum_{k=0}^\infty \frac{240}{(4k+5)(4k+6)(4k+7)(4k+9)(4k+10)(4k+11)}=\frac{22}{7}-\pi$$
https://math.stackexchange.com/a/1657416/134791
(UPDATE Peter Bala New series for old functions https://oeis.org/A002117/a002117.pdf, 2009, formula 5.1)
Equivalently,
$$\sum_{k=1}^\infty \fra... | {
"language": "en",
"url": "https://mathoverflow.net/questions/67384",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "139",
"answer_count": 6,
"answer_id": 3
} |
Number of Permutations? Edit: This is a modest rephrasing of the question as originally stated below the fold: for $n \geq 3$, let $\sigma \in S_n$ be a fixed-point-free permutation. How many fixed-point-free permutations $\tau$ are there such that $\sigma \tau^{-1}$ is also fixed-point free? As the original post shows... | Let me copy here an answer from Russian forum dxdy.ru that I obtained using the approach outlined in my paper.
Two given rows of a $3\times N$ matrix define a permutation of order $N$. Let $c_i$ ($i=1,2,\dots,N$) be the number of cycles of length $i$ in this permutation (in particular, $c_1$ is the number of fixed poin... | {
"language": "en",
"url": "https://mathoverflow.net/questions/144899",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 5,
"answer_id": 4
} |
Convexity of a certain sublevel set Consider the polynomial of degree $4$ in variable $r$
$$ r^4 + (x^2 + y^2)\ r^2 - 2 x y\ r + x^2 y^2 $$
The discriminant of this polynomial in $r$ is the following expression
(obtained using Mathematica)
$$ \Delta := 16 x^2 y^2 \left( -x^6 + x^8 - 27 x^2 y^2 + 33 x^4 y^2 - 4 x^6 y^2... | You can parametrize the zero locus of $\Delta$ (other than the origin) in the first quadrant by
$$
(x,y) = \left(\frac{(3{-}t)\sqrt{(3{+}t)(1{-}t)}}{8},
\frac{(3{+}t)\sqrt{(3{-}t)(1{+}t)}}{8}\right)
\qquad\qquad -1\le t\le 1
$$
and then compute that the curvature is
$$
\frac{(x'y''-y'x'')}{((x')^2+(y'... | {
"language": "en",
"url": "https://mathoverflow.net/questions/177559",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 2,
"answer_id": 1
} |
Prime divisors of $p^n+1$ Let $p$ be a rational prime and $n$ be a positive integer.
It can be easily deduced from Zsigmondy's theorem that $p^n+1$ has a prime divisor greater than $2n$ except when $(p,n)=(2,3)$ or $(2^k-1,1)$ for some positive integer $k$. Hence we know that there exists an odd prime divisor of $p^n+... | $x^n+1$ factors over $\mathbb{Z}[x]$ unless $n$ is a power of two.
For $n=15$ the factorization is $ (x + 1) \cdot (x^{2} - x + 1) \cdot (x^{4} - x^{3} + x^{2} - x + 1) \cdot (x^{8} + x^{7} - x^{5} - x^{4} - x^{3} + x + 1)$.
The factors $(p^{2} - p + 1)$ and $(p^{4} - p^{3} + p^{2} - p + 1)$ are odd and with congruence... | {
"language": "en",
"url": "https://mathoverflow.net/questions/191473",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
The elliptic curve for $x_1^9+x_2^9+\dots+x_6^9 = y_1^9+y_2^9+\dots+y_6^9$ I. Theorem: "If there are $a,b,c,d,e,f$ such that,
$$a+b+c = d+e+f\tag1$$
$$a^2+b^2+c^2 = d^2+e^2+f^2\tag2$$
$$3u^3-3uv+w=-def\tag3$$
where $u=a+b+c,\; v = ab+ac+bc,\;w = abc$, then,
$$(a + u)^k + (b + u)^k + (c + u)^k + (d - u)^k + (e - u)^k + ... | For the equation,
\begin{equation*}
y^2=(-7x^3-21c_1x+c_2)(7c_1x+c_2)
\end{equation*}
where $c_1=(n^2+3)/8$ and $c_2=(n^3-9n)/2$, and assuming $n$ is rational, we have a rational point
$(0,c_2)$, so the quartic is birationally equivalent to an elliptic curve.
Using an ancient Ms-Dos version of Derive, it is easy to use... | {
"language": "en",
"url": "https://mathoverflow.net/questions/193787",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Common roots of polynomial and its derivative Suppose $f$ is a uni-variate polynomial of degree at most $2k-1$ for some integer $k\geq1$. Let $f^{(m)}$ denote the $m$-th derivative of $f$. If $f$ and $f^{(m)}$ have $k$ distinct common roots then, Is it true that $f$ has to be a zero polynomial? Here $m<k$ is a positiv... | Assume $a,b,c \in \mathbb{R}$ solve
$$2(a^3+b^3+c^3)-3(a^2b+ab^2+b^2c+bc^2+a^2c+ac^2)+12abc=0,$$
e.g. $(a,b,c)=(-1,1,3)$. Then
$$
\begin{eqnarray}
f(x)&:=&(x-a)(x-b)(x-c)(3x^2-2(a+b+c)x+3(ab+bc+ca)-2(a^2+b^2+c^2))\\
&=&3x^5-5(a+b+c)x^4+10(ab+bc+ca)x^3\\
&&+(2(a^3+b^3+c^3)-3(a^2b+ab^2+\dots)-18abc)x^2+\dots
\end{eqnarr... | {
"language": "en",
"url": "https://mathoverflow.net/questions/203601",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
Is this a rational function? Is $$\sum_{n=1}^{\infty} \frac{z^n}{2^n-1} \in \mathbb{C}(z)\ ?$$
In a slightly different vein, given a sequence of real numbers $\{a_n\}_{n=0}^\infty$, what are some necessary and sufficient conditions for $\sum a_nz^n$ to be in $\mathbb{C}(z)$ with all poles simple?
| $\sum a_n z^n$ is a rational function iff $a_n$ is a sum of polynomials times exponentials. This is a straightforward corollary of partial fraction decomposition. So, suppose $\frac{1}{2^n - 1}$ can be expressed as such a sum. Taking $n \to \infty$ shows that the largest $r$, in absolute value, such that $r^n$ appears ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/208071",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 7,
"answer_id": 3
} |
rational numbers and triangular numbers This question is an offshoot of Ratio of triangular numbers. Suppose $ka(a+1)=nb(b+1)$, where $k,n >1$ are relative prime integers, and $a,b \geq 0$ are integers. Which $k,n$ pairs have no solution other than the trivial one $a = 0, b = 0$?
(Checked for $k,n<21$. Found no solutio... | There should always be solutions unless $kn$ is a square. The equation is
equivalent to
$$k (2a+1)^2 - n (2b+1)^2 = k - n.$$
Let $(x_0, y_0)$ be
the fundamental solution of the Pell equation $x^2 - 4 k n y^2 = 1$.
Then
$$ a = \frac{x_0-1}{2} + n y_0, \quad b = \frac{x_0-1}{2} + k y_0 $$
give you a solution.
For $k = ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/224445",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How many integer solutions of $a^2+b^2=c^2+d^2+n$ are there? Are there any references to study the integer solutions (existence and how many) of Diophantine equations like $a^2+b^2=c^2+d^2+2$, $a^2+b^2=c^2+d^2+3$, $a^2+b^2=c^2+d^2+5$...? Actually, I can prove that there are integer solutions (a,b,c,d) for any integers ... | Here's an elementary way to see that there are always plenty of solutions. Find $r$ and $s$ such that $r+s=n$ and neither $r$ nor $s$ is twice an odd number --- for large $n$, there will be many ways to do this. Then, there always exist $a,b,c,d$ such that $a^2-c^2=r$ and $b^2-d^2=s$, whence $a^2+b^2=c^2+d^2+n$.
| {
"language": "en",
"url": "https://mathoverflow.net/questions/239675",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
A conjecture harmonic numbers I will outlay a few observations applying to the harmonic numbers that may be interesting to prove (if it hasn't already been proven).
From the Online Encyclopedia of Positive Integers we have:
$a(n)$ is the number of permutations $p$ of $\{1,\ldots,n\}$ such that the minimum number of blo... | We can use the characterization by Christie. Let $\pi \in S_n$. Add a fixed point $0$ to $\pi$, and let $c$ be the cycle $(0, 1, \ldots, n)$. Then the smallest number of block interchanges to sort $\pi$ is equal to $\frac{n + 1 - t}{2}$, where $t$ is the number of cycles in decomposition of $c \pi^{-1} c^{-1} \pi$. Whe... | {
"language": "en",
"url": "https://mathoverflow.net/questions/376935",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 2,
"answer_id": 1
} |
The Diophantine equation $ (xz+1)(yz+1)=az^{3}+1$ has no solutions in positive integers with $z > a^2+2a$ Problem. Let $a$ be a positive integer that is not a perfect cube. Show that the Diophantine equation $(xz+1)(yz+1)=az^{3}+1$ has no solutions in positive integers $x, y, z$ with $z > a^{2}+2a$.
If this result is ... | The given equation $(xz+1)(yz+1)=az^3+1$ can be rewritten as $az^2-xyz-(x+y)=0$. We shall show that for any solution $(x,y,z)$, we have $z \le a^2+2a. \ $
Note that $z \ | \ x+y$, therefore $z \le x+y. \ $ Treating $x, y$ as constants, the only positive solution for $z \ $ is \begin{equation} z = \frac{xy+\sqrt{x^2y... | {
"language": "en",
"url": "https://mathoverflow.net/questions/386989",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Show that these matrices are invertible for all $p>3$ I am working on a paper which will extend a result in my thesis and have boiled one problem down to the following: show that the symmetric matrix $M_p$, whose definition follows, is invertible for all odd primes $p$. Letting $p>3$ be prime and $\ell = \frac{p-1}{2}$... | Experimentally, we have the following formula for $p$ prime:
$$\det(M_p)=(-1)^{(p^2-1)/8}(2p)^{(p-3)/2}h_p^-\;,$$
where $h_p^-$ is the minus part of the class number of the $p$-th cyclotomic
field, itself essentially equal to a product of $\chi$-Bernoulli numbers.
I have not tried to prove this, but since there are man... | {
"language": "en",
"url": "https://mathoverflow.net/questions/417800",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
"answer_count": 2,
"answer_id": 1
} |
Prove positivity of rational functions We say a rational function $F(z)$ is positive if the coefficients of its Maclaurin expansion, in the variable $z$, are non-negative.
In this context, let
$$F_r(z):=\frac{1 - 2z + z^r - (1 - z)^r}{(1 - z)^{r - 1}(1 - 2z)}.$$
Is the following true? Note: $F_2(z)=0$ and $F_3(z)$ is e... | Notice that
$$F_r(z) = \frac{1}{(1-z)^{r-1}} - \sum_{k=0}^{r-1} \left(\frac{z}{1-z}\right)^k$$
and therefore for $r\geq 4$ and $n\geq 1$, we have
\begin{split}
[x^n]\ F_r(z) &= \binom{n+r-2}{r-2} - \sum_{k=1}^{r-1} \binom{n-1}{k-1} \\
& = \binom{n+r-2}{r-2} - \binom{n-1}{r-2} - \binom{n-1}{r-3} - \sum_{k=1}^{r-3} \bino... | {
"language": "en",
"url": "https://mathoverflow.net/questions/422218",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Solve this sextic I'm working with the expression $m = 32an^6+96an^5+120an^4+80an^3+28an^2+4bn^2+4an+4bn+2a+2c
$. What exactly is the closed radical form of this, if one were to write $n$ in terms of $m$? There's no general formula for sixth powers but I want to know if this particular class can be given in radicals fo... | Mathematica finds a closed-form expression for the solutions $n$ as a function of $m$ of the equation
$$m = 32an^6+96an^5+120an^4+80an^3+28an^2+4bn^2+4an+4bn+2a+2c.$$
The expressions for general $a,b,c$ are lengthy. By way of example, for $a=1$, $b=2$, $c=-3$ the two real solutions are
$$n=-\tfrac{1}{2}\pm\tfrac{1}{2}\... | {
"language": "en",
"url": "https://mathoverflow.net/questions/425287",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Some Log integrals related to Gamma value Two years ago I evaluated some integrals related to $\Gamma(1/4)$.
First example:
$$(1)\hspace{.2cm}\int_{0}^{1}\frac{\sqrt{x}\log{(1+\sqrt{1+x})}}{\sqrt{1-x^2}} dx=\pi-\frac{\sqrt {2}\pi^{5/2}+4\sqrt{2}\pi^{3/2}}{2\Gamma{(1/4)^{2}}}.$$
The proof I have is based on the followin... | I was hoping to finish it but I am stuuck with a last integral. Then, this is just a comment.
Consider
$$I(a)=\int_{0}^{1}\frac{\sqrt{x}\log{(a+\sqrt{1+x})}}{\sqrt{1-x^2}}\, dx$$
$$I'(a)=\int_{0}^{1} \frac{\sqrt{x}}{\sqrt{1-x^2} \left(a+\sqrt{x+1}\right)}\,dx$$
$$I'(a)=\pi-\pi \sqrt{1+\frac{1}{a^2-2}}-2 a K(-1)+2 a\, ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/437979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Equality of the sum of powers Hi everyone, I got a problem when proving lemmas for some combinatorial problems,
and it is a question about integers.
Let
$\sum_{k=1}^m a_k^t = \sum_{k=1}^n b_k^t$
be an equation,
where $m, n, t, a_i, b_i$ are positive integers, and
$a_i \neq a_j$ for all $i, j$,
$b_i \neq b_j$ for all ... | An even harder problem than $t>2$ and $n=m$ is the Prouhet–Tarry–Escott problem. Now I leave it to you and google to find lots of examples ;-)
http://en.wikipedia.org/wiki/Prouhet-Tarry-Escott_problem
| {
"language": "en",
"url": "https://mathoverflow.net/questions/16764",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 0
} |
Evaluation of the following series... $S = 1/(2\times3) + 1/(5\times6) + 1/(7\times8) + 1/(10\times11) + ... $ EDIT, Will Jagy, December 8, 2010: to anyone considering working on this, please first see http://mathoverflow.tqft.net/discussion/817/could-a-few-moderators-please-remove-one-of-my-questions/#Item_9
which ... | As you are interested in $ \zeta(3) $ you might prefer this variant of your construction. This is also a small part of what Pietro Majer would have done in the direction indicated by Charles Matthews.
Define $ f(x)$ for $ | x | \leq 1 $ by
$$
f(x) = \frac{x^4}{2 \cdot 3 \cdot 4} + \frac{x^7}{5 \cdot 6 \cdot 7} ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/26035",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 8,
"answer_id": 3
} |
An optimization problem The following problem optimization problem arose in a project I am working on with a student. I would like to minimize the quantity:
$$ M=\frac{1}{12} + \int_0^\frac{1}{2} \left( \tfrac{1}{2}-x \right)^2 Q(x)^2 \ dx - 4 \left[ \int_0^\frac{1}{2} \left( \tfrac{1}{2}-x \right) Q(x) \ dx\right]^2 ... | I tried calculus of variations. It's been a while since I've done calculus of variations, so I could be messing it up completely. Also, this isn't completely rigorous. There are ways of making calculus of variations rigorous, but I don't know them, and they're a lot harder than just doing calculations.
By my comment ab... | {
"language": "en",
"url": "https://mathoverflow.net/questions/51339",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Counting card distributions when cards are duplicated If we have a deck of 48 different cards and 4 players each get 12 cards, it is well known how to calculate the number of possible distributions: take fac(48) and divide 4 times by fac(12).
In a german card came (Doppelkopf) there are 24 different card types, but 2 c... | Label the players as 1,2,3, and 4. We first count the total possible number of hands that 1 can be dealt. Let $a$ be the number of singletons 1 has and let $b$ be the number of duplicates 1 has. This yields a total number of
$$\sum \binom{24}{a} \binom{24-a}{b}$$ possibilities, where the sum ranges over all
$a$ and $... | {
"language": "en",
"url": "https://mathoverflow.net/questions/55752",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
} |
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.
| Of course for the equation $X^2+Y^2=Z^2+t$
There is a particular solution:
$X=1\pm{b}$
$Y=\frac{(b^2-t\pm{2b})}{2}$
$Z=\frac{(b^2+2-t\pm{2b})}{2}$
But interessuet is another solution:
$X^2+Y^2=Z^2+1$
If you use the solution of Pell's equation: $p^2-2s^2=\pm1$
Making formula has the form:
$X=2s(p+s)L+p^2+2ps+2s^2=aL+c$
... | {
"language": "en",
"url": "https://mathoverflow.net/questions/65957",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 4
} |
Generalized tic-tac-toe We begin with $2n+1$ cards, each with a distinct number from $-n$ to $+n$ on it, face up in between the two players of the game. The players take turns selecting a card and keeping it. The first player to collect three cards that sum to zero wins the game. If the cards are exhausted and neith... | First player wins for $n$ at least five. First turn, name $0$. They name a number, say $-a$. Choose two numbers $b$ and $c$ such that neither $b$, $c$, nor $b+c=a$. Then name $b$, forcing them to name $-b$, then $c$, forcing them to name $-c$, then $-b-c$, winning. You can always choose two such numbers, since each pos... | {
"language": "en",
"url": "https://mathoverflow.net/questions/103787",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Determinant of non-symmetric sum of matrices Given three real, symmetric matrices $A\succ0$ and $B$, $C⪰ 0$.
How can it be shown that:
$$\det(A^2+AB+AC) \leq \det(A^2 +BA +AC+BC) ? \qquad (\star)$$
Where $A^2$ is symmetric and positive definite. Eigenvalues of $BA$, $AC$, and $BC$ are all $> 0$, but symmetry is lost.
T... | None of the conjectured inequalities hold.
This answer contains three counterexamples. The first one is to $(\star)$, while the second and third ones (below the line) refer to previous inequalities conjectured by the OP.
\begin{equation*}
A=\begin{bmatrix} 5 & 5\\\\ 5 & 5\end{bmatrix},\quad
B=\begin{bmatrix} 8 &4 \... | {
"language": "en",
"url": "https://mathoverflow.net/questions/131953",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
An entropy inequality
Let $X,Y$ be probability measures on $\{1,2,\dots,n\}$, and set $K=\sum_i\sqrt{X(i)Y(i)}$ so that $Z:=\frac{1}{K}\sqrt{XY}$ is also a probability measure on $\{1,2,\dots,n\}$. How can we prove the inequality
$$H(X)+H(Y)\geq 2K^2 H(Z),$$ where $H(X)=-\sum_{i=1}^n X(i)\log X(i)$ is the entropy fun... | Remark: This is to give an alternative proof of the inequality in fedja's nice answer:
$$
(1+a)\log(1+b)+(1+b)\log(1+a)\ge 2(1+c)\log(1+c)
$$
where $a, b, c>0$ with $ab=c^2$.
Proof.
WLOG, assume that $a \le b$.
Let $x = \frac{a}{c} \in (0, 1]$.
It suffices to prove that
$$f(x) := (1 + cx)\ln(1 + c/x) + (1 + c/x)\ln(1 +... | {
"language": "en",
"url": "https://mathoverflow.net/questions/138275",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "36",
"answer_count": 3,
"answer_id": 2
} |
Decomposition of $SU(3)$ representation $6\times 15$ into irreducibles? The 6 and 15 dimensional representations of $SU(3)$ are irreducible. The 90 dimensional tensor product representation $6\times 15$ decomposes into a sum of irreducible representations. What factors occur and with what multiplicity?
Note: by 6 I mea... | This question is borderline between what is on topic and what isn't; if you want to do a number of computations like this you should pick up a book on representation theory. My standard recommendations for $SL_n$ rep theory are Chapter 8 of Fulton's Young Tableaux or Appendix II (by Fomin) in Stanley's Enumerative Comb... | {
"language": "en",
"url": "https://mathoverflow.net/questions/146294",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Binomial coefficient identity It seems to be nontrivial (to me) to show that the following identity holds:
$$ \binom {m+n}{n} \sum_{k=0}^m \binom {m}{k} \frac {n(-1)^k}{n+k} = 1. $$
This quantity is related to the volume of the certain polytope.
| This can be proved by induction on $m$ (for all $n$). That is, we want to prove by induction on $m \geq 0$ that
$$
\binom{m+n}{n} \sum_{k=0}^m \binom{m}{k} \frac{n(-1)^k}{n+k} = 1
$$
for all $n \geq 1$. When $m = 0$ the left side is 1 for all $n$. If the above equation holds for $m$, then we want to show
$$
\binom{m+... | {
"language": "en",
"url": "https://mathoverflow.net/questions/193611",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 9,
"answer_id": 2
} |
What is known about primes of the form $x^2-2y^2$? David Cox's book Primes of The Form: $x^2+ny^2$ does a great job proving and motivating a lot of results for $n>0$. I was unable to find anything for negative numbers, let alone the case I am interested in, $n=-2$.
What is the reason for this? Maybe I am missing somet... | As $2 = x^2 -2y^2$ for $x = 2$ and $y=1$ we fix an odd prime number $p$, and
Claim: There exist integers $x,y$ such that $p = x^2 -2y^2$ if and only if $p \equiv \pm1 \mod 8$.
First if $p = x^2 -2y^2$ for some $x,y \in \mathbb{Z}$ then observing that the squares mod $8$ are $0,1,4$, we conclude that $p$ can only be $0... | {
"language": "en",
"url": "https://mathoverflow.net/questions/197918",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 3,
"answer_id": 2
} |
Action of upper triangular matrices Let $M,N$ be two $n\times m$ matrices with $n\leq m$ and coefficients in an algebraically closed field of characteristic zero $K$, both of full rank $n$.
Do there exist two upper triangular matrices $A\in SL(n)$ and $B\in SL(m)$ such that $A\cdot M \cdot B^{T} = \lambda N$ for $\lam... | I think the answer to your question is negative. Consider for instance
$$M = \begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}, \quad N = \begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}$$
Assume that there exist
$$A = \begin{pmatrix}
a_{11} & a_{12} \\
0 & a_{22}
\end{pmatrix}, \quad B = \begin{pmatrix}
b_{11} & b_{12} \\ ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/285250",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
On the polynomial $\sum_{k=0}^n\binom{n}{k}(-1)^kX^{k(n-k)}$ Let $n = 2m$ be an even integer and let $F_n(X)$ be the polynomial $$F_n(X):=\sum_{k=0}^n\binom{n}{k}(-1)^kX^{k(n-k)}.$$ I observed (but cannot prove) that the polynomial $F_n$ is always divisible by $(1-X)^m$, and the quotient $F_n(X)/(1-X)^m$ always has pos... | Here is a proof for the positivity of coefficients. Suppose that
$$F_{2m}\,(x)=2(1-x)^m\sum_ka_{m,k}\,x^k,$$
where $a_{m,k}=0$ if $k<0$ or $k>m^2-m$.
We shall show that $a_{m,k}>0$ for all other $k$.
First of all, a negative sign was missed in Joe Silverman's deduction from the third last step. The correct formula is ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/296465",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "33",
"answer_count": 6,
"answer_id": 5
} |
Dividing a cake between $n-1$, $n$, or $n+1$ guests
A housewife is waiting for guests and has prepared a cake. She doesn't know how many guests will come, but it will be $n-1$, $n$, or $n+1$.
What is the minimal number $f(n)$ of pieces the cake should be cut to make it possible to divide between guests equally?
Fo... | $f(7)=15$.
$f(7)\ge15$ follows from a comment of Fedor Petrov on the original question, so it suffices to find a way to cut the cake into $15$ pieces so as to serve $6$, $7$, or $8$ guests.
Let the size of the cake be $168$ (so that all the following computations involve only whole numbers). Let the $15$ pieces be of... | {
"language": "en",
"url": "https://mathoverflow.net/questions/330683",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "35",
"answer_count": 3,
"answer_id": 0
} |
Factorizing a bivariate polynomial I have a bivariate polynomial for each $n=0,1,2...$
$$
f_n(x,y)=\sum _{k=0}^n \frac{(-1)^k}{2 k+1} \binom{n}{k} \left(x ^2-y ^2\right)^{2 n-2 k}\left([y ( x^2 -1) +x(1 -y^2 )]^{2 k+1}\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad-[y ( x^2 -1) -x(1 -y^2 )]^{2 k+1}\right)
$$
... | So,
\begin{split}
f_n(\xi,\eta) &= (\xi^2-\eta^2)^{2n+1} \left( F\big(\frac{\eta ( \xi^2 -1) +\xi(1 -\eta^2)}{\xi^2-\eta^2}\big) - F\big(\frac{\eta ( \xi^2 -1) -\xi(1 -\eta^2)}{\xi^2-\eta^2}\big)\right) \\
&= (\xi^2-\eta^2)^{2n+1} \left( F\big(\frac{\eta\xi + 1}{\xi+\eta}\big) - F\big(\frac{\eta\xi - 1}{\xi-\eta}\big)\... | {
"language": "en",
"url": "https://mathoverflow.net/questions/339390",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Computing the integral $\int_{-1}^1 dx \, |x| J_0(\alpha \sqrt{1 - x^2}) P_\ell(x)$ I would like to compute the following integral:
$$
I_\ell(\alpha) := \int_{-1}^1 dx \, |x| J_0(\alpha \sqrt{1 - x^2}) P_\ell(x)
\tag{1}
\label{1}
$$
where $\alpha \geq 0$, $J_0$ is the zeroth-order Bessel function of the first kind, $P_... | Thanks to the comment by Johannes, the solution can indeed be obtained by using the following identities:
\begin{equation}
P_\ell(z)
=
\frac{1}{2^\ell}
\sum\limits_{k=0}^{\left\lfloor \frac{\ell}{2}\right\rfloor}
(-1)^k
\begin{pmatrix} \ell \\ k \end{pmatrix}
\begin{pmatrix} 2\ell - 2k \\ \ell \end{pmatrix}
z^{\ell - 2... | {
"language": "en",
"url": "https://mathoverflow.net/questions/365861",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Generating functions of Collatz iterates? Let $C(n) = n/2$ if $n$ is even and $3n+1$ otherwise be the Collatz function.
We look at the generating function $f_n(x) = \sum_{k=0}^\infty C^{(k)}(n)x^k$ of the iterates of the Collatz function.
The Collatz conjecture is then equivalent to: For all $n$:
$$f_n(x) = p_n(x) + x^... | Without assuming the Collatz conjecture, it can be shown that the generating functions satsify certain polynomial equations:
Observe that for all $n$:
$$f_{C(n)}(x) = \frac{f_n(x)-n}{x}$$
hence:
$$f_{C^{(2)}(n)}(x) = \frac{f_{C(n)}(x)-C(n)}{x}$$
Solving for $x$ and equating the two identities and letting $x_k:=f_{C^{(k... | {
"language": "en",
"url": "https://mathoverflow.net/questions/375044",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 1,
"answer_id": 0
} |
Minimal polynomial in $\mathbb Z[x]$ of seventh degree with given roots I am looking for a seventh degree polynomial with integer coefficients, which has the following roots.
$$x_1=2\left(\cos\frac{2\pi}{43}+\cos\frac{12\pi}{43}+\cos\frac{14\pi}{43}\right),$$
$$x_2=2\left(\cos\frac{6\pi}{43}+\cos\frac{36\pi}{43}+\cos\f... | In SageMath, you can enter the following:
U.<zeta> = CyclotomicField(43)
P.<x> = PolynomialRing(U)
def c(j): # cos(j * pi / 43)
return (zeta ** j + zeta ** (-j))/2
x1 = 2*(c(2) + c(12) + c(14))
x2 = 2*(c(6) + c(36) + c(42))
x3 = 2*(c(18) + c(22) + c(40))
x4 = 2*(c(20) + c(32) + c(34))
x5 = 2*(c(10) + c(16) + c(2... | {
"language": "en",
"url": "https://mathoverflow.net/questions/375278",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Characteristic polynomial of a matrix related to pairs of elements generating $\mathbb{Z}/n\mathbb{Z}$ Fix $n\geq 2$. Let $A$ be the matrix whose rows and columns
are indexed by pairs $(a,b)\in \mathbb{Z}/n\mathbb{Z}$ such
that $a,b$ generate $\mathbb{Z}/n\mathbb{Z}$ (the number of
such pairs is $\phi(n)\psi(n)$, where... | The corank is $\phi(n)\psi(n)/6$ for all $n>3$. In particular, your computations imply that the matrix is not diagonalizable for $n=6,7,9$.
This follows from the fact that all nonzero entries are accumulated in the $6\times 6$ blocks with rows indexed by
$$
(a,b),\; (a-b,a),\; (-b,a-b),\; (-a,-b),\; (b-a,a), \; (b,b-... | {
"language": "en",
"url": "https://mathoverflow.net/questions/376940",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
} |
An infinite series that converges to $\frac{\sqrt{3}\pi}{24}$ Can you prove or disprove the following claim:
Claim:
$$\frac{\sqrt{3} \pi}{24}=\displaystyle\sum_{n=0}^{\infty}\frac{1}{(6n+1)(6n+5)}$$
The SageMath cell that demonstrates this claim can be found here.
| Write the sum as $S_{o}=\sum_{n=1}^{\infty} \frac{1}{(3(2n+1)-2)(3(2n+1)+2)}=\sum_{n=\text{odd}} \frac{1}{(3n)^2-(2)^2}$.
Hence, $S_o=S-S_e$. Where, for $S$ , $n$ are all natural numbers in the above expression and for $S_e$, $n$ takes only positive even numbers.
Hence, from the identity $\frac{\pi\text{cot}(x)}{x}=\fr... | {
"language": "en",
"url": "https://mathoverflow.net/questions/400819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 0
} |
Positive integer solutions to the diophantine equation $(xz+1)(yz+1)=z^4+z^3 +z^2 +z+1$ Let $P(z) = z^4 +z^3 +z^2 +z+1$ where $z$ is a positive integer.
While working with the diophantine equation $(xz+1)(yz+1)=P(z)$, I was able to construct a seemingly infinite and complete solution set to the diophantine equation in ... | "OP" wants to know of a numerical solution which does not satisfy his parametric solution. It is given below.
$(x,y,z)=(138,49331482518,2609165)$
| {
"language": "en",
"url": "https://mathoverflow.net/questions/403542",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
Can $y^2-4$ be a divisor of $x^3-x^2-2 x+1$? Do there exist integers $x$ and $y$ such that $\frac{x^3-x^2-2 x+1}{y^2-4}$ is an integer?
In other words, can any integer representable as $x^3-x^2-2 x+1$ have any divisor representable as $y^2-4$?
This is the simplest non-trivial example of my earlier question Integer poin... | No. The roots of $x^3 - x^2 - 2x + 1$ are $-(\zeta + \zeta^{-1})$ where $\zeta$ is a 7th root of unity; this soon implies [see below] that any prime factor is either $7$ or $\pm 1 \bmod 7$, and thus that all factors of $x^3 - x^2 - 2x + 1$ are congruent to $0$ or $\pm 1 \bmod 7$. In particular it is not possible for ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/417804",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
} |
Convergence rate of a sequence Suppose $x_0=x_1=1$, define $y_k=x_k+\frac{1}{2}(x_k-x_{k-1})$ and $x_{k+1}=y_k-\eta y_k^3$ where $\eta\in(0,1/8)$. If we know $x_k\to 0$ as $k\to\infty$. How to show that $x_k=\Theta(1/\sqrt{k})$?
It suffices to show that $x_k^{-2}=\Theta(k)$. By Taylor expansion, we have
$$x_{k+1}^{-2}... | Suppose that we already know that $x_k>0\forall k$. Clearly $(x_k)$ and $(y_k)$ are decreasing sequences which converge to $0$. Then we can prove that $b_k:=\frac{x_{k}}{x_{k-1}}\to 1$. To do it note first that $b_k\in[0,1]\forall k$ and
$$b_{k+1}=\frac{y_k-\eta y_k^3}{x_k}=\frac{x_k+\frac{1}{2}(x_k-x_{k-1})-\eta y_k^3... | {
"language": "en",
"url": "https://mathoverflow.net/questions/432266",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Equality of the sum of powers Hi everyone, I got a problem when proving lemmas for some combinatorial problems,
and it is a question about integers.
Let
$\sum_{k=1}^m a_k^t = \sum_{k=1}^n b_k^t$
be an equation,
where $m, n, t, a_i, b_i$ are positive integers, and
$a_i \neq a_j$ for all $i, j$,
$b_i \neq b_j$ for all ... | One set of solutions for t = 3 is the class of numbers known as Taxicab Numbers, named after the number of a taxicab G. H. Hardy took, 1729, that Ramanujan mentioned was equal to 13 + 123 = 93 + 103. This particular example fails, as |10 - 9| = 1 < 2, but there are other Taxicab numbers, such as:
1673 + 4363 = 2283 + ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/16764",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 1
} |
'Sign matrices'-(-1,+1) square matrices My question arises from a discussion on an answer given by Maurizio Monge here.I do not know if there is a known terminology for such matrices. By "sign matrices," I mean square matrices whose entries are in ${-1,+1}$.
For instance,
$\begin{bmatrix}
1 &-1 \\
-1& -1
\end{bmat... | It is not part of the question but I consider it useful to give the result for the $M_3$ case( which I found using a $C$++ program). There are 512 rows and the matrix below is $M_3$. However, I am not completely certain since the compiler I used might have rounded off the numbers while multiplying the matrices though t... | {
"language": "en",
"url": "https://mathoverflow.net/questions/40451",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
A Diophantine equation Let $a,b$ be integers, $a>b\ge 1$,
$a^2(a+1)$ be divisible by $b$, and
$3a^2$ be divisible by $b$.
Let us consider the following expression:
$\frac {1+3a+3a^2+a^2(a-b+1)/b}
{1+\frac{3}{a}+\frac{3}{a^2}+\frac{b}{a^2(a-b+1)}}$.
This fraction is always integer if $b=1$. For $b>1$, I know only one ... | If we substitute $f= \frac{a-b+1}{ab}$ and subtract $a^3$ the question becomes; when is this an integer:
$$\frac{a^3 \times (f-1) \times (a^3 + \frac{1}{f})}{(a+1)^3 + \frac{1}{f} -1}$$
Note that $b=1$ implies that $f=1$ and this quotient is zero.
Using Maple I found that $a=14$, $b=12$ is a second example which satisf... | {
"language": "en",
"url": "https://mathoverflow.net/questions/66654",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$ - \sum_{n=1}^\infty \frac{(-1)^n}{2^n-1} =? \sum_{n=1}^\infty \frac{1}{2^n+1}$ Numerical evidence suggests:
$$ - \sum_{n=1}^\infty \frac{(-1)^n}{2^n-1} =? \sum_{n=1}^\infty \frac{1}{2^n+1} \approx 0.764499780348444 $$
Couldn't find cancellation via rearrangement.
For the second series WA found closed form.
Is the e... | This is just a formal proof using idea of Loïc Teyssier that appeared in the comments. We just rewrite both sides of the equation as double sums (using geometric series) and notice that they are equal after changing the order of summation.
Since $$\frac{q}{1-q} = \sum_{k=1}^\infty q^k,$$ we have for the left hand side
... | {
"language": "en",
"url": "https://mathoverflow.net/questions/143592",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
On class numbers $h(-d)$ and the diophantine equation $x^2+dy^2 = 2^{2+h(-d)}$ Given fundamental discriminant $d \equiv -1 \bmod 8$ such that the quadratic imaginary number field $\mathbb{Q}(\sqrt{-d})$ has odd class number $h(-d)$. Is it true that one can always solve the diophantine equation,
$$x^2+dy^2 = 2^{2+h(-d)}... | Or, put the cheesy way: with $d = p \equiv 7 \pmod 8,$ there is a primitive positive quadratic form with coefficients $$ \langle 2, 1 , \frac{p+1}{8} \rangle. $$ As a result, with class number $h,$ by repeated Gauss composition we know the principal form
$$ \langle 1, 1 , \frac{p+1}{4} \rangle $$ integrally represent... | {
"language": "en",
"url": "https://mathoverflow.net/questions/148624",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Sets of squares representing all squares up to $n^2$ Let $S_n=\{1,2,\ldots,n\}$ be natural numbers up to $n$.
Say that a subset $S \subseteq S_n$
square-represents $S_n^2$ if every
square $1^2,2^2,\ldots,n^2$ can be represented by adding or subtracting
at most one copy of squares of elements of $S$.
Example. For $n=7$,... | Here is a little example following Jeremy Rouse's construction. Let $A$ be the $a_i$ terms
and $B$ the $b_i$ terms.
$$A=\{1,2,3,4,5,6,8,11,16\}$$
$$B=\{0,0,0,0,42,78,142,263,519\}$$
For example
$$a_9 = \lfloor \sqrt{b_8} \rfloor = \lfloor \sqrt{263} \rfloor = 16 \;,$$
and
$$b_9=b_8+a^2_9 = 263+16^2 = 519 \;.$$
The mis... | {
"language": "en",
"url": "https://mathoverflow.net/questions/191185",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Can a block matrix with at least 3 zero blocks of different size on the diagonal and 1's everywhere else have only integer eigenvalues? Let $M=\begin{pmatrix}
\begin{array}{cccccccc}
0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\
0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\
1 & 1 & 0 & 0 & 0 & 1 & 1 &1\\
1 & 1 & 0 & 0 & 0 & 1 & 1 &1\\
1 & 1 & ... | You can have a look at Eigenvalues of complete multipartite graphs. One example with integer spectrum is $K_{2,2,2,8,8\ }.$
| {
"language": "en",
"url": "https://mathoverflow.net/questions/225251",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Integral quaternary forms and theta functions The following question arises when I attempt to understand the modular parameterization of the elliptic curve $$E:y^2-y=x^3-x$$
In Mazur-Swinnerton-Dyer and Zagier's construction, a theta function associated with a positive definite quadratic form is induced:
$$\theta(q)=\s... | The answer is true, using the following construction.
Let $B$ be the quaternion algebra of discriminant $p$ and let $O$ be a maximal order with an element $x$ satisfying $x^2 = -p$. The reduced norm is a quadratic form on $O$, with positive definite Gram matrix $A$ of determinant $p^2$. The matrix $A^{-1}$ then represe... | {
"language": "en",
"url": "https://mathoverflow.net/questions/231770",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Solution to a Diophantine equation Find all the non-trivial integer solutions to the equation
$$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=4.$$
| There is one idea. To search for the solution of the equation.
$$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=q$$
If we know any solution $(a,b,c)$ of this equation. Then it is possible to find another $(a_2, b_2, c_2)$. Make such a change.
$$y=(a+b+2c)(q(a+b)-c)-(a+b)^2-(a+c)(b+c)$$
$$z=(a+2b+c)(2b-qa-(q-1)c)+(b+2a+c)(2... | {
"language": "en",
"url": "https://mathoverflow.net/questions/264754",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 1
} |
How to deduce an equation from this 3 Diophantine equations with 5 variables? I have three equations:
${m \choose 2} + nk = {x \choose 2}$
${n \choose 2} + mk = {y \choose 2}$
$x + y = m + n + k$
$m, n, k, x, y$ are natural numbers. I want to deduce from this 3 equations either $x = y$ or $m = n$. From where I got the... | The first two equalities imply $x>m$ and $y>n$ so one can substitute $x=m+X$, $y=n+Y$ and $k=X+Y$, with still $X,Y \in \mathbb N$:
${X \choose 2}=nX+nY-mX\tag{1}$
${Y \choose 2}=mX+mY-nY\tag{2}$
From (1) follows: $\quad m=n+n\frac{Y}{X}-\frac{1}{X}{X \choose 2}$,
then eliminate $m$ from (2): $\quad {Y \choose 2}+{X \ch... | {
"language": "en",
"url": "https://mathoverflow.net/questions/277766",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Cycle index of $S_n \times S_m$ Given $n$, $m$ and $k$, I would like to evaluate cycle index of $S_n \times S_m$ for $c_1 = c_2 = ... = c_{nm} = k$. What is the fastest known algorithm to calculate it? For this particular case, is there any polynomial time algorithm? I know that $Z(S_n) = \frac{1}{n}\sum_{i=1}^na_iZ(S_... | $Z(S_n \times S_m)$ evaluated at $c_1 = c_2 = ... = c_{nm} = k$ equals
$$\sum_{i_1+2i_2+\dots+ni_n=n}\frac{1}{1^{i_1} i_1!\cdots n^{i_n} i_n!} \sum_{j_1+2j_2+\dots+mj_m=m} \frac{1}{1^{j_1} j_1!\cdots m^{j_m} j_m!}\cdot
k^{\sum_{p=1}^n\sum_{q=1}^m \gcd(p,q)i_p j_q}.$$
There are $p(n)\cdot p(m)$ terms in this double sum.... | {
"language": "en",
"url": "https://mathoverflow.net/questions/291012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Minimal polynomial in $\mathbb Z[x]$ of seventh degree with given roots I am looking for a seventh degree polynomial with integer coefficients, which has the following roots.
$$x_1=2\left(\cos\frac{2\pi}{43}+\cos\frac{12\pi}{43}+\cos\frac{14\pi}{43}\right),$$
$$x_2=2\left(\cos\frac{6\pi}{43}+\cos\frac{36\pi}{43}+\cos\f... | By PARI / GP I get
$x^7 + x^6 - 18*x^5 - 35*x^4 + 38*x^3 + 104*x^2 + 7*x - 49$ :
K = nfinit (subst(polcyclo(43),x,y))
w = Mod(y,K.pol)
f0(k) = (w^k + 1/w^k)
f(k1,k2,k3) = f0(k1) + f0(k2) + f0(k3)
v = [f(1,6,7),f(3,18,21),f(9,11,20),f(10,16,17),f(5,8,13),f(4,15,19),f(2,12,14)]
/*
=
[x^7 + x^6 - 18x^5 - 35x^4 + 38x^3 + 1... | {
"language": "en",
"url": "https://mathoverflow.net/questions/375278",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Can a product of Cohn matrices over the Eisenstein integers with non-zero, non-unit coefficients be a Cohn matrix? For $k > 1$, is it possible that $\begin{pmatrix} a_1 & 1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} a_2 & 1 \\ -1 & 0 \end{pmatrix}\ldots \begin{pmatrix} a_k & 1 \\ -1 & 0 \end{pmatrix} = \pm \begin{pmatrix} ... | To my surprise, not only is there a solution for some $b$, there is actually a very simple infinite family of solutions for every $b$. Let $\omega = \frac{1 + \sqrt{-3}}{2}$. Then
$\begin{pmatrix} a_0 + a_1\omega & 1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} 2 -\omega & 1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} 1 + \omega ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/376328",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
How to show a $3\times3$ matrix has three distinct eigenvalues? Here is a question I heared from others:
Given four distinct positive real numbers $a_1,a_2,a_3,a_4$ and set $$a:=\sqrt{\sum_{1\leq i\leq 4}a_i^2}$$
$A=(x_{i,j})_{1\leq i\leq3,1\leq j\leq4}$ is a $3\times4$-matrix specified by $$ x_{i,j}=a_i\delta_{i,j}+a_... | This question is from this year's Alibaba mathematics competition (qualifying round, which is finished 2 days ago), and here's my solution that could be wrong (I also participated in the competition and this is the solution I submitted). I tried to solve the problem geometrically to avoid tons of computations.
First, ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/393053",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 2
} |
Subtraction-free identities that hold for rings but not for semirings? Here is a concrete, if seemingly unmotivated, aspect of the question I am interested in:
Question 1. Let $a$ and $b$ be two elements of a (noncommutative) semiring $R$ such that $1+a^3$ and $1+b^3$ and $\left(1+b\right)\left(1+a\right)$ are inverti... | The answer to your first question is yes (which was very surprising to me, to be honest). I have no idea whether the second question also has a positive answer. (By the way, don't let the work below fool you. This took me an entire week of serious computations to discover the main idea.)
We will assume $(1+a^3)u=1$ ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/398544",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "26",
"answer_count": 3,
"answer_id": 1
} |
An infinite series that converges to $\frac{\sqrt{3}\pi}{24}$ Can you prove or disprove the following claim:
Claim:
$$\frac{\sqrt{3} \pi}{24}=\displaystyle\sum_{n=0}^{\infty}\frac{1}{(6n+1)(6n+5)}$$
The SageMath cell that demonstrates this claim can be found here.
| Here is an elementary proof. We rewrite the series as
$$\frac{1}{4}\int_0^1\frac{1-x^4}{1-x^6}\,dx=\frac{1}{8}\int_0^1\frac{dx}{1-x+x^2}+\frac{1}{8}\int_0^1\frac{dx}{1+x+x^2}.$$
It is straightforward to show that
\begin{align*}
\int_0^1\frac{dx}{1-x+x^2}&=\frac{2\pi}{3\sqrt{3}},\\
\int_0^1\frac{dx}{1+x+x^2}&=\frac{\pi}... | {
"language": "en",
"url": "https://mathoverflow.net/questions/400819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 1
} |
$2n \times 2n$ matrices with entries in $\{1, 0, -1\}$ with exactly $n$ zeroes in each row and each column with orthogonal rows and orthogonal columns I am interested in answering the following question:
Question
For a given $n$, does there exist a $2n \times 2n$ matrix with entries in $\{1, 0, -1\}$ having orthogonal ... | Your first conjecture was proven by Nate in the comments.
Your second conjecture is also true - there is no such matrix for $n=3$. If we just look at which entries are nonzero in each row, because any two rows are orthogonal, they must share an even number of nonzero entries, i.e. either share $0$ entries or $2$ entrie... | {
"language": "en",
"url": "https://mathoverflow.net/questions/421691",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 0
} |
Equality of the sum of powers Hi everyone, I got a problem when proving lemmas for some combinatorial problems,
and it is a question about integers.
Let
$\sum_{k=1}^m a_k^t = \sum_{k=1}^n b_k^t$
be an equation,
where $m, n, t, a_i, b_i$ are positive integers, and
$a_i \neq a_j$ for all $i, j$,
$b_i \neq b_j$ for all ... | For any $t$, if $m$ is sufficiently large relative to $t$, and $n$ is any positive integer (possibly equal to $m$), then the circle method proves that there exists an infinite sequence of increasingly large solutions such that the ratios between the $a_1,\ldots,b_n$ approach any real positive ratios you want (assuming ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/16764",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 2
} |
Pythagorean 5-tuples What is the solution of the equation $x^2+y^2+z^2+t^2=w^2$ in polynomials over C ("Pythagorean 5-tuples")?
There are simple formulas describing Pythagorean n-tuples for n=3,4,6:
*
*n=3. The formula for solutions of $x^2+y^2=z^2$ [4]:
$x=d(p^2−q^2)$,
$y=2dpq$,
$z=d(p^2+q^2),$
where p,q,d are ... | In fact, the equation:
$X^2+Y^2+Z^2+R^2=W^2$
Solutions look like this:
$X=2psabk^2+a^2k^2s^2-ckabs^2-abk^2s^2$
$Y=2psabk^2+a^2k^2s^2-ckabs^2+2abk^2s^2$
$Z=2psabk^2+a^2k^2s^2+2ckabs^2-abk^2s^2$
$R=2p^2b^2k^2+c^2b^2s^2+b^2k^2s^2-ckb^2s^2-a^2k^2s^2+2psabk^2$
$W=2p^2b^2k^2+2psabk^2+c^2b^2s^2+b^2k^2s^2-ckb^2s^2+2a^2k^2s^2$
... | {
"language": "en",
"url": "https://mathoverflow.net/questions/62820",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
"answer_count": 6,
"answer_id": 5
} |
Integral of the product of Normal density and cdf I am struggling with an integral pretty similar to one already resolved in MO (link: Integration of the product of pdf & cdf of normal distribution ). I will reproduce the calculus bellow for the sake of clarity, but I want to stress the fact that my computatons are ess... | When you do the change of variable, you are doing:
\begin{align}
\left[y\longmapsto f\frac{\sqrt{1+B^2}}{B}-\frac{A}{B\sqrt{1+B^2}}\Longrightarrow df=\frac{B}{\sqrt{1+B^2}}\,dy\right]
\end{align}
.. which is correct. However, we need to apply the same change of variable to the bounds, ie to $-\infty$ and $+\infty$. Giv... | {
"language": "en",
"url": "https://mathoverflow.net/questions/127086",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
Solutions to $\binom{n}{5} = 2 \binom{m}{5}$ In Finite Mathematics by Lial et al. (10th ed.), problem 8.3.34 says:
On National Public Radio, the Weekend Edition program posed the
following probability problem: Given a certain number of balls, of
which some are blue, pick 5 at random. The probability that all 5 ar... | This isn’t a complete answer, but the problem “reduces” to finding the finitely many rational points on a certain genus 2 hyperelliptic curve. This is often possible by a technique
involving a reduction to finding the rational points on a finite set of rank 0 elliptic curves—see for example “Towers of 2-covers of hyper... | {
"language": "en",
"url": "https://mathoverflow.net/questions/128036",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "26",
"answer_count": 1,
"answer_id": 0
} |
Symmetric sums and Representations of SO(3) I had tried to help someone on math.StackExchange to prove the identity:
$$ (1-Tr(A))^2+\sum_{1\le i\le j\le 3}(a_{ij}-a_{ji})^2=4$$
I guess you could argue the left hand side is independent of basis. Then we can diagonalize. But I couldn't come up with an invariant way of ... | Using the parametrization
$$
A= \begin{pmatrix} a^2+b^2-c^2-d^2&2bc-2ad &2bd+2ac \cr
2bc+2ad &a^2-b^2+c^2-d^2&2cd-2ab \cr
2bd-2ac &2cd+2ab &a^2-b^2-c^2+d^2\\ \end{pmatrix},
$$
which comes from quaternions of unit length we have
$$
(1-tr (A))^2-\frac{1}{2} tr ((A-A^t)^2)-4=
$$
$$
(9a^2 + b^2 + c^2 + d^2 + 3)(a^2 +... | {
"language": "en",
"url": "https://mathoverflow.net/questions/131614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Pythagorean triples related to non-isometric equidistant plane quadruples QUESTION Do there exist integers $u\ x\ A\ B$ such that $x\ne 0$, and the following two equalities hold:
*
*$ x^2 + (x-u)^2\ =\ A^2$
*$ x^2 + (x+u)^2\ =\ B^2$
?
REMARK I have a family of pairs of quadruples $S\ T\subseteq\math... | It's not even possible for the product of $x^2 + (x-u)^2$ and $x^2 + (x+u)^2$ to be a square unless $x=0$ or $u=0$: that product is $4x^4+u^4$, and
the elliptic curve $4x^4+u^4 = y^2$ is isomorphic to $Y^2 = X^3 - X$,
for which Fermat already proved that the obvious rational points are
the only ones. For your applic... | {
"language": "en",
"url": "https://mathoverflow.net/questions/135050",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Four-Square Theorem for Negative Coefficient What integers are not in the range of $a^2+b^2+c^2-x^2$ (for all integer combinations of a, b, c, and x)? This form is similar to that of Lagrange's Four-Square Theorem, for which the answer would be "none". The generalization by Ramanujan only seems to cover non-negative ... | With indefinite forms, it is possible for ternary forms to be universal. Indeed, all are known. References are given in Modern Elementary Theory of Numbers by Leonard Eugene Dickson, (1939). With any integer $M$ and any odd $N,$ they are equivalent to (by an invertible linear change of variables) one of four:
$$ xy ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/138282",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How to prove that the odd continued fraction approximants of ln(1+X) are upper bounds? The odd order continued fraction approximants for $\ln(1+X)$ are
$$X,\quad \frac{X^2+6X}{4X+6,}\quad \frac{X^3+21X^2+30X}{9X^2+36X+30,}\quad \dots.$$
In "Some bounds for the logarithmic function", Flemming Topsøe remarks that they ar... | The main result of the paper you link is that
$$\log(1+x) = \frac{x}{1+} \frac{x}{2+} \frac{x}{3+} \frac{2^2 x}{4+} \frac{2^2 x}{5+} \cdots$$
and what you want to know is that truncating this formula at an odd number of steps provides an upper bound. (I am using the standard shorthand $\frac{a}{b+} \frac{c}{d+} \cdots$... | {
"language": "en",
"url": "https://mathoverflow.net/questions/152625",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Hypergeometric sum specific value How to show?
$${}_2F_1(1,1;\frac{1}{2}, \frac{1}{2}) = 2 + \frac{\pi}{2} $$
It numerically is very close, came up when evaluating:
$$ \frac{1}{1} + \frac{1 \times 2}{1 \times 3} + \frac{1 \times 2 \times 3}{1 \times 3 \times 5} + \frac{1 \times 2 \times 3 \times 4}{1 \times 3 \times 5 ... | This is equivalent to showing that $\displaystyle\sum_{n=2}^\infty\frac{n!}{(2n-1)!!}=\frac\pi2$ , which, after multiplying both the
numerator and the denominator with $(2n)!!=2^n\,n!$, and taking into account that $\dfrac{(2n)!}{n!^2}=$
$=\displaystyle{2n\choose n}$, can be rewritten as $\displaystyle\sum_{n=2}^\inf... | {
"language": "en",
"url": "https://mathoverflow.net/questions/163212",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
PDF of the product of normal and Cauchy distributions I am having trouble in finding out the resulting PDF of the product of normal and Cauchy distributions. It turns out that we have a general formula for calculating the PDF of product of two random distribution but the integral is not converging . Also, I tried using... | Assuming the normal is centered with variance $s$ and the Cauchy distribution has parameters $a, b$, combining this Wikipedia page and Mathematica gives
$$
\text{ConditionalExpression}\left[\frac{i \left(\frac{e^{-\frac{z^2}{2 s^2 (a-i b)^2}}
\left(2 \pi \text{erfi}\left(\frac{\left| z\right| }{\sqrt{2} (a s-i b s)... | {
"language": "en",
"url": "https://mathoverflow.net/questions/192144",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Binomial coefficient identity It seems to be nontrivial (to me) to show that the following identity holds:
$$ \binom {m+n}{n} \sum_{k=0}^m \binom {m}{k} \frac {n(-1)^k}{n+k} = 1. $$
This quantity is related to the volume of the certain polytope.
| This is a specialization at $x = n$ of the rational function identity
$$
\frac{1}{x(x+1)\cdots(x+m)} = \frac{1}{m!}\sum_{k=0}^m \binom{m}{k}\frac{(-1)^k}{x+k},
$$
and this identity could be proved either by partial fractions with unknown coefficients and limits to find the coefficients (see my comment on Todd Trimble'... | {
"language": "en",
"url": "https://mathoverflow.net/questions/193611",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 9,
"answer_id": 4
} |
Negative coefficient in an almost cyclotomic polynomial Let $a,b,c,d$ be four prime numbers. We set the polynomial :
$$P(X)=\frac{(1-X^{abc})(1-X^{abd})(1-X^{acd})(1-X^{bcd})(1-X^a)(1-X^b)(1-X^c)(1-X^d)}{(1-X)^2(1-X^{ab})(1-X^{ac})(1-X^{ad})(1-X^{bc})(1-X^{bd})(1-X^{cd})}$$
By numerical tests, i see that $P(X)$ always ... | Suppose that $a<b<c<d$. We show that the coefficient of $X^c$ or of $X^{b+c-1}$ of $P(X)$ is negative. In order to do so, it suffices to work in the power series ring $\mathbb Q[[X]]$ modulo $X^{b+c}$. Note that $ac>b+c$ and so on, hence
\begin{equation}
P(X)\equiv\frac{(1-X^a)(1-X^b)(1-X^c)(1-X^d)}{(1-X)^2(1-X^{ab})}\... | {
"language": "en",
"url": "https://mathoverflow.net/questions/216322",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Integer solution For every prime $p$, does there exists integers $x_1$, $x_2$ and $x_3$ ($0\leq x_1, x_2, x_3 \leq \lfloor cp^{1/3}\rfloor$ and $c$ is some large constant) such that $\frac{p-1}{2}-\lfloor 2cp^{1/3} \rfloor \leq f(x_1,x_2,x_3) \leq \frac{p-1}{2}$, where, $f(x_1,x_2,x_3)=x_1+x_2+x_3+2(x_1x_2+x_2x_3+x_3x_... | I doubt that the lower bound $\frac{p-1}{2} - 2cp^{1/3}$ holds for all $p$. Here is a proof for the weaker bound $\frac{p-1}{2} - cp^{1/2}$.
First of all, the inequality $\frac{p-1}{2}-cp^{1/2} \leq f(x_1,x_2,x_3) \leq \frac{p-1}{2}$ is essentially equivalent to
$$p- O(p^{1/2}) \leq (2x_1+1)(2x_2+1)(2x_3+1) \leq p.$$
... | {
"language": "en",
"url": "https://mathoverflow.net/questions/217829",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Can a block matrix with at least 3 zero blocks of different size on the diagonal and 1's everywhere else have only integer eigenvalues? Let $M=\begin{pmatrix}
\begin{array}{cccccccc}
0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\
0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\
1 & 1 & 0 & 0 & 0 & 1 & 1 &1\\
1 & 1 & 0 & 0 & 0 & 1 & 1 &1\\
1 & 1 & ... | It is not a complete solution, but can be such of them by some calculations.
You can assign a graph to the matrice $M$ in each case and analyze them. If all-zero blocks have equal size, their size $t$ must be the divisors of $8$. So, all possible cases are $1$, $2$, $4$ and $8$. If $t=1$ then you have the complete grap... | {
"language": "en",
"url": "https://mathoverflow.net/questions/225251",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
An irresistible inequality The following occurred while working on some research project. Since the methods of proof I used were lengthy, I wish to see a skillful or insightful approach (perhaps even conceptual). Anyhow, here it is. Let
$$f(x)=\left(\frac{x}{e^x-1}\right)^2 + \left(\frac{x+1}{e^{x+1}+1}\right)^2.$$
Ca... | I do not know wether this helps you or not, but you may do as follows.
Denote $f(x)=(e^x-1)/x$. Note that $f'(x)=\frac{fe^x(x-1+e^{-x})}{x(e^x-1)}>0$, so
$f$ is a positive increasing function.
Lemma. The function $1/f=x/(e^x-1)$ is a (positive decreasing) convex function.
Proof.
$$
(1/f)''=\frac{e^x(2+x-(2-x)e^x)}{(e... | {
"language": "en",
"url": "https://mathoverflow.net/questions/243939",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Is there always one integer between these two rational numbers? It appears that for each integer $k\geq2$, there is always one integer $c$ that satisfies the inequalities below. Can this be proved?
$$\frac{3^k-2^k}{2^k-1}<c\leq \frac{3^k-1}{2^k}.$$
Note that for $k\geq2$ the lower bound is always a proper fraction and ... | We show that $\left\lfloor\frac{3^n-1}{2^n-1}\right\rfloor$ and $\left\lfloor \left(\frac{3}{2} \right)^n\right\rfloor$ have a common floor when $\frac{3^n-1}{2^n-1}-\left(\frac{3}{2} \right)^n =$ $ \left\{\frac{3^n-1}{2^n-1}\right\}-\left\{\left(\frac{3}{2} \right)^n\right\} = \delta(n),$ where $\left\{\cdot\right\}$ ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/280192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 2
} |
Primality test for specific class of Proth numbers Can you provide a proof or a counterexample for the following claim :
Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$
Let $N=k\cdot 2^n+1$ such that $n>2$ , $0< k <2^n$ and
$\begin{cases} k \equiv 1,7 \pmod{30}... | Your criterion is equivalent to:
$$(4+\sqrt{15})^{k2^{n-1}}\equiv (4+\sqrt{15})^{\frac{N-1}{2}}\equiv -1 (\bmod N \mathbb{Z}[\sqrt{15}]).$$
The point is that $P_m(8)=(4+\sqrt{15})^m+(4-\sqrt{15})^m$. Moreover, $S_i=(4+\sqrt{15})^{k2^i}+(4-\sqrt{15})^{k2^i}$, which one may prove by induction, using the fact that $x\ma... | {
"language": "en",
"url": "https://mathoverflow.net/questions/307601",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 0
} |
Oddity of generalized Catalan numbers: Part I The famous (classical) Catalan numbers $C_{1,n}=\frac1{n+1}\binom{2n}n$ satisfy the following well-known arithmetic property:
$$\text{$C_{1,n}$ is odd iff $n=2^j-1$ for some $j$}.\tag1$$
Consider the "second generation" of Catalan numbers $C_{2,n}$ which can be found on OEI... | The answer to QUESTION 1 is Yes.
The following proof simplifies my suggestion in the comments.
According to the OEIS entry, the shifted generating function
$$
g(x) = \sum_{n=0}^\infty C_{2,n} x^{n+1} = x + 2x^2 + 8x^3 + 39x^4 + \cdots
$$
satisfies $g^4 - 2g^2 + g = x$. Reducing $\bmod 2$ gives $g^4 + g = x.$
Therefo... | {
"language": "en",
"url": "https://mathoverflow.net/questions/324816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 0
} |
Bounds on the mills ratio How do I show the following bounds on the mills ratio :
$\frac{1}{x}- \frac{1}{x^3} < \frac{1-\Phi(x)}{\phi(x)} < \frac{1}{x}- \frac{1}{x^3} +\frac{3}{x^5} \ \ \ \ \ \ \ $ for $ \ \ \ x>0$ where $\Phi()$ is the CDF of the Normal distribution , and $\phi()$ is the density function of the N... | Here's a sketch and a link for how I prove it. Let
$$ f(x) = - \left( \frac{1}{x} - \frac{1}{x^3} + \frac{3}{x^5}\right) \phi(x) .$$
Now show (lemma): $\frac{df}{dx} = \left(1 + \frac{15}{x^6}\right)\phi(x)$.
(To prove this, use that $\frac{d\phi}{dx} = - x \phi(x)$, the product rule, and some cancellation.)
Now supp... | {
"language": "en",
"url": "https://mathoverflow.net/questions/330054",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Two versions of Sylvester identity MathWorld presents the following two versions of Sylvester's determinant identity, relating to an $n\times n$ matrix $\mathbb{A}$:
First:
$$
|\mathbb{A}||A_{r\,s,p\,q}| = |A_{r,p}||A_{s,q}| - |A_{r,q}| |A_{s,p}|
$$
where $r$ and $s$ ($p$ and $q$) are sets that indicate which rows (col... | For the special case that $r,s,p,q$ are single elements, it is shown in these notes (page 7) how the first identity (known as the Desnanot-Jacobi identity) follows from the second identity.
Apply the second identity to the matrix
we thus arrive at the first identity, illustrated graphically as
source
Update: Since I... | {
"language": "en",
"url": "https://mathoverflow.net/questions/370336",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Minimal polynomial in $\mathbb Z[x]$ of seventh degree with given roots I am looking for a seventh degree polynomial with integer coefficients, which has the following roots.
$$x_1=2\left(\cos\frac{2\pi}{43}+\cos\frac{12\pi}{43}+\cos\frac{14\pi}{43}\right),$$
$$x_2=2\left(\cos\frac{6\pi}{43}+\cos\frac{36\pi}{43}+\cos\f... | I also used PARI/GP with the following program:
z1 = Mod(z, (z^43-1)/(z-1));
e(n) = lift(Mod(3,43)^n);
c(n) = z1^n + z1^-n;
r(n) = c(1*n) + c(6*n) + c(7*n);
p = prod(n=1,7, x - r(e(n)));
lift(p)
with the resulting output
z^7+z^6-18*z^5-35*z^4+38*z^3+104*z^2+7*z-49
A simpler program with complex numbers is
z1=exp(2*Pi... | {
"language": "en",
"url": "https://mathoverflow.net/questions/375278",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Evaluations of three series involving binomial coefficients Question. How to prove the following three identities?
\begin{align}\sum_{k=1}^\infty\frac1{k(-2)^k\binom{2k}k}\left(\frac1{k+1}+\ldots+\frac1{2k}\right)=\frac{\log^22}3-\frac{\pi^2}{36},\tag{1}
\end{align}
\begin{align}\sum_{k=1}^\infty\frac1{k2^k\binom{3k}k}... | Here is a proof of the first identity. The others can probably be done similarly.
We have
$$\frac1{k\binom{2k}{k}}=\frac12 B(k,k)=\frac12 \int_0^1 t^{k-1}(1-t)^{k-1}\,dt,$$
$$\frac{1}{k+1}+\cdots+\frac{1}{2k} = \int_0^1 \frac{1-x^{2k}}{1+x}\, dx$$
and
$$\sum_{k=1}^\infty \frac{1}{(-2)^k}t^{k-1}(1-t)^{k-1}(1-x^{2k}) = -... | {
"language": "en",
"url": "https://mathoverflow.net/questions/383967",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Finding all proper divisors of $a_3z^3 +a_2z^2 +a_1z+1$ of the form $xz+1$ Let $n=a_3z^3+a_2z^2+a_1z+1$ where $a_1<z, \ a_2<z, \ 1 \le a_3<z, z>1$ are non negative integers. To obtain proper divisors of $n$ of the form $xz+1$, one may perform trial divisions $xz+1 \ | \ n$, for all $xz+1 \le \sqrt n$. Trial division h... | Such an extension is highly unlikely to exist. Already in the simple case of $z=2$, it's equivalent to just factoring a given odd integer $n$, which is a famous hard problem.
| {
"language": "en",
"url": "https://mathoverflow.net/questions/393596",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Divisibility of (finite) power sum of integers Consider the power sum
$$S_a(b)=1^{2b}+2^{2b}+\cdots+(3a-2)^{2b}.$$
Let $\nu_3(x)$ denote the $3$-adic valuation of $x$.
QUESTION 1. (milder) Is this true?
$$\nu_3\left(\frac{S_a(b)}{S_a(1)}\right)=0.$$
QUESTION 2. Is this true? $\nu_3(S_a(b))=\nu_3(2a-1)$.
| Notice that
$$2S_a(b) \equiv 1^{2b} + 2^{2b} + \dots + (6a-4)^{2b} \pmod{6a-3}.$$
From Faulhaber's formula, we have
$$1^{2b} + 2^{2b} + \dots + (6a-4)^{2b} \equiv B_{2b} (6a-3) \pmod{6a-3}.$$
It follows that
$$S_a(b) \equiv \frac32B_{2b} (2a-1)\pmod{3(2a-1)},$$
where $\nu_3(\frac32 B_{2b})=0$ by von Staudt–Clausen theo... | {
"language": "en",
"url": "https://mathoverflow.net/questions/397146",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Primality test for $\frac{(10 \cdot 2^n)^m-1}{10 \cdot 2^n-1} - 2$ and $\frac{(10 \cdot 2^n)^m+1}{10 \cdot 2^n+1} - 2$ Here is what I observed:
For $\frac{(10 \cdot 2^n)^m-1}{10 \cdot 2^n-1} - 2$ :
Let $N$ = $\frac{(10 \cdot 2^n)^m-1}{10 \cdot 2^n-1} - 2$ :
when $m$ is a number $m \ge 3$ and $n \ge 0$.
Let the sequence... | It's often the case with such tests that the "only if" part is more or less easy to prove, while the "if" part is inaccessible for proving or disproving. Below I prove the "only if" part, ie. assuming that $N$ is prime.
First notice that Chebyshev polynomials appear here just for efficient computation of Lucas number
$... | {
"language": "en",
"url": "https://mathoverflow.net/questions/422588",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Surface fitting with convexity requirement Hi all,
Consider a cloud of points in 3D space (x,y,z). The data is well-behaved, once plotted the surface looks like some sort of spheroid. I assumed a form for the fitting function f(x,y,z) = c1 x^2 + c2 y^2 + c3 z^2 + c4 x^2y^2 + ..etc
The coefficients were obtained using a... | For your apparent purpose, in dimension $n$ it is convenient to begin with a homogeneous polynomial of total degree $2n$ with all individual exponents even. Then, for translates and rotations, all sorts of lower degree and odd exponent terms may show up.
In $\mathbb R^2,$ a rounded version of an ordinary square is
$$ ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/73538",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Hypergeometric sum specific value How to show?
$${}_2F_1(1,1;\frac{1}{2}, \frac{1}{2}) = 2 + \frac{\pi}{2} $$
It numerically is very close, came up when evaluating:
$$ \frac{1}{1} + \frac{1 \times 2}{1 \times 3} + \frac{1 \times 2 \times 3}{1 \times 3 \times 5} + \frac{1 \times 2 \times 3 \times 4}{1 \times 3 \times 5 ... | You may use the general formula from Brychkov, Prudnikov, Marichev, Integral and Series, Vol.3:
$$
_{2}F_{1}(1,1;\frac{1}{2};x)=(1-x)^{-1}\left(1+\frac{\sqrt{x}\arcsin{\sqrt{x}}}{\sqrt{1-x}}\right)
$$
and put in it $x=\frac{1}{2}$.
| {
"language": "en",
"url": "https://mathoverflow.net/questions/163212",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
When is $(q^k-1)/(q-1)$ a perfect square? Let $q$ be a prime power and $k>1$ a positive integer. For what values of $k$ and $q$ is the number $(q^k-1)/(q-1)$ a perfect square, that is the square of another integer? Is the number of such perfect squares finite?
Note that $(q^k-1)/(q-1)$ is the number of points in a fin... | The equation
$$ \frac{x^k-1}{x-1}=y^m$$
is known as the Nagell-Ljunggren equation. It is conjectured that for $x\geq 2$, $y\geq 2$, $k\geq 3$, $m\geq 2$, the only solutions are
$$ \frac{3^5-1}{3-1}=11^2,\qquad \frac{7^4-1}{7-1}=20^2,\qquad \frac{18^3-1}{18-1}=7^3.$$
For $m=2$, the equation was solved by Ljunggren (Nor... | {
"language": "en",
"url": "https://mathoverflow.net/questions/177952",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 1,
"answer_id": 0
} |
Infinite limit of ratio of nth degree polynomials The Problem
I have two recursively defined polynomials (skip to the bottom for background and motivation if you care about that) that represent the numerator and denominator of a factor and I want to find the limit of that factor as n goes to infinity.
$$n_0 = d_0 = 1$... | The explanation for your observations is that you are dealing with a self-adjoint difference equation in disguise. Let me make a transformation along these line, though I won't analyze it through to the end.
If we let $Y_n=(n_n,d_n)^t$, then this satisfies
$$
Y_n = \begin{pmatrix} -1 & x \\ -1 & x-1 \end{pmatrix} Y_{n-... | {
"language": "en",
"url": "https://mathoverflow.net/questions/249549",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 2,
"answer_id": 0
} |
Quasi-concavity of $f(x)=\frac{1}{x+1} \int_0^x \log \left(1+\frac{1}{x+1+t} \right)~dt$ I want to prove that function
\begin{equation}
f(x)=\frac{1}{x+1} \int\limits_0^x \log \left(1+\frac{1}{x+1+t} \right)~dt
\end{equation}
is quasi-concave. One approach is to obtain the closed form of the integral (provided below) ... | The derivative is: ${d f(x) \over d x} = \frac{-\log (x+1)+\log (x+2)+\log \left(\frac{x+1}{2 x+1}\right)}{(x+1)^2}$, which is:
and hence is both positive and negative, as Mark L. Stone wrote.
| {
"language": "en",
"url": "https://mathoverflow.net/questions/303704",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Want to prove an inequality I want to show that
$9*\left[\frac{xy}{x+y}-q(1-q)\right]-12*[xy-q(1-q)]+(1-q-x)^{3}+(x+y)^{3}+(q-y)^{3}-1\geq0$ where
$0<q<1$
$0<x<1-q$
$0<y<q$
$(x+y)\left[1+max\{\frac{1-q}{y},\frac{q}{x}\}\right]\leq3$
I play with it numerically. It is right. But don't know how to prove it analytically... | Remark: Actually, the inequality is quite easy. The trick is to take $x, y$ as parameters.
Denote the expression by $f(q)$.
It suffices to prove that $f(q) \ge 0$ provided that
\begin{align*}
&x, y > 0,\\
&x + y < 1,\\
&3y > (x + y)^2,\\
&3x > (x + y)^2, \\
&y < q < 1 - x, \\
&1 + y - \frac{3y}{... | {
"language": "en",
"url": "https://mathoverflow.net/questions/330587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
Estimation of Hypergeometric function ${_3F_2}$ Is there any way to estimate the following function, which is a result of sum of ratios of Gamma functions?
$$
{_3F_2}\begingroup
\renewcommand*{\arraystretch}
% your pmatrix expression
\left[
\begin{array}{c@{}c}
\begin{array}{c}
-q, \frac{M}{2}, \frac{1}{2}+\f... | Here are explicit forms of
$$f(q,n,m)={_3F_2}\begingroup
\renewcommand*{\arraystretch}
% your pmatrix expression
\left[
\begin{array}{c@{}c}
\begin{array}{c}
-q, \frac{m}{2}, \frac{1}{2}+\frac{m}{2}\\
\frac{1}{2}, -q-\frac{n-m}{2}+1
\end{array} ;& 1
\end{array}\r... | {
"language": "en",
"url": "https://mathoverflow.net/questions/344116",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
The exact constant in a bound on ratios of Gamma functions The answer to another question (Upper bound of the fraction of Gamma functions) gave an asymptotic upper bound for an expression with Gamma functions:
$$\left(\frac{\Gamma(a+b)}{a\Gamma(a)\Gamma(b)}\right)^{1/a}\!\leq \,C\,\frac{a+b}a, \forall a,b\geq\frac12$$
... | The optimal $C$ is $\mathrm{e}$.
Proof:
We have
$$\ln C \ge \ln a - \ln(a + b)
+ \frac{\ln \Gamma(a + b)
-\ln a - \ln\Gamma(a) - \ln\Gamma(b)}{a}.$$
Let
$$F(a, b) := \ln a - \ln(a + b)
+ \frac{\ln \Gamma(a + b)
-\ln a - \ln\Gamma(a) - \ln\Gamma(b)}{a}.$$
We have
$$\frac{\partial F}{\partial b}
= - \frac{1}{a+b}... | {
"language": "en",
"url": "https://mathoverflow.net/questions/352806",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
A question on an identity relating certain sums of Harmonic numbers In the description of this question, it was established that \begin{align} \sum_{n=2}^{\infty} (\zeta(n)^{2}-1) &=
\frac{7}{4} - \zeta(2) + 2 \sum_{m=1}^{\infty} \frac{H_{m-1- \frac{1}{m}} - H_{- \frac{1}{m}} - H_{m-1} }{m} \qquad(1) \end{align}
In t... | Q3: The first identity (1) is not correct, it should read
\begin{align} \sum_{n=2}^{\infty}(\zeta(n)^{2} -1) &= \frac{7}{4} - \zeta(2) + 2\sum_{m=2}^{\infty} \frac{H_{m-1-\frac{1}{m}} - H_{-\frac{1}{m}} - H_{m-1}}{m} . \end{align}
The final identity (*) then becomes
$$
\sum_{m=2}^{\infty} \frac{H_{-\frac{1}{m}}}{m} = \... | {
"language": "en",
"url": "https://mathoverflow.net/questions/373499",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Divisibility condition implies $a_1=\dotsb=a_k$? Let $a_1, a_2, \dotsc, a_k$ be $k$ positive integers, $k\ge2$. For all $n\ge n_0$ there is a positive integer $f(n)$ such that $n$ and $f(n)$ are relatively prime and $a_{1}^{f(n)}+\dotsb+a_{k}^{f(n)}$ is a multiple of $a_{1}^n+\dotsb+a_{k}^n$.
Is it true that $a_1=a_2=\... | Here's a a tweak of Seva's idea that gives a counterexample. Note that if $r$ is odd, then $2^{n}+1$ divides $2^{rn} + 1$.
Let $k = 6$, $a_{1} = 1$, $a_{2} = a_{3} = a_{4} = 2$, $a_{5} = a_{6} = 4$. Then $a_{1}^{n} + \cdots + a_{6}^{n} = 1 + 3 \cdot 2^{n} + 2 \cdot 4^{n} = (1+2^{n})(1+2^{n+1})$.
If $n$ is any positive ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/378733",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 1
} |
Finding all proper divisors of $a_3z^3 +a_2z^2 +a_1z+1$ of the form $xz+1$ Let $n=a_3z^3+a_2z^2+a_1z+1$ where $a_1<z, \ a_2<z, \ 1 \le a_3<z, z>1$ are non negative integers. To obtain proper divisors of $n$ of the form $xz+1$, one may perform trial divisions $xz+1 \ | \ n$, for all $xz+1 \le \sqrt n$. Trial division h... | An extension to the case when $n<z^5$ with some restrictions on $x$ and $y$:
Let $n=a_4z^4+a_3z^3+a_2z^2+a_1z+1$, $a_i < z $, $a_4>0, z>1$. We are looking for positive integes $x$ and $y$ such that $(xz+1)(yz+1)=n$. We add a restriction on $x$ and $y$; If $x=x_k \cdot z^k + \cdots + x_0$ and $y=y_t \cdot z^t + \cdots ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/393596",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
EM-wave equation in matter from Lagrangian Note
I am not sure if this post is of relevance for this platform, but I already asked the question in Physics Stack Exchange and in Mathematics Stack Exchange without success.
Setup
Let's suppose a homogeneous dielectric with a (spatially) local dielectric response function $... | Expanding on Carlo Beenakker's comment, one can't just expect to substitute in a complex dielectric function to properly describe absorption. Rather, the relevant question is how to structure a Lagrangean such as to generate the desired damping term in the equation of motion. For example, to generate linear damping in ... | {
"language": "en",
"url": "https://mathoverflow.net/questions/404380",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
What are the properties of this set of infinite matrices and operations on them? Consider infinite matrices of the form
$$\left(
\begin{array}{ccccc}
a_0 & a_1 & a_2 & a_3 & . \\
0 & a_0 & a_1 & a_2 & . \\
0 & 0 & a_0 & a_1 & . \\
0 & 0 & 0 & a_0 & . \\
. & . & . & . & . \\
\end{array}
\right)$$
The elements on ea... | If the matrices have entries from a (unital) ring $R$ then the set of such matrices is isomorphic to $R[[x]]$, the ring of formal power series over $R$. To see this, observe that the map sending the infinite matrix with $a_0 = 0$, $a_1 = 1$ and $a_k = 0$ for $k \ge 2$ to $x$ is a ring isomorphism.
This also answers the... | {
"language": "en",
"url": "https://mathoverflow.net/questions/404518",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Are there finitely many primes $x$ such that for a fixed odd prime $p$, $n=x^{p-1}+x^{p-2}+\dotsb + x+1$ is composite and $x \mid \phi(n)$? Let
\begin{equation} n =x^{p-1}+x^{p-2}+\dotsb + x+1
\end{equation} where $x$ and $p$ are odd primes.
If $p$ is set to $5$, it appears $x=5$ is the only prime $x$ such that $n$ is ... | Here's a proof of Conjecture 1 for the case $p=5$. The proof depends on the truth of the following overwhelmingly true unproven result :
Let
\begin{equation} P(x) = x^4 +x^3 +x^2 +x+1.
\end{equation}
Then all positive integers $x$ such that $P(x) $ has a proper divisor congruent to 1 modulo $x$ are given as follows :
... | {
"language": "en",
"url": "https://mathoverflow.net/questions/405529",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
$2n \times 2n$ matrices with entries in $\{1, 0, -1\}$ with exactly $n$ zeroes in each row and each column with orthogonal rows and orthogonal columns I am interested in answering the following question:
Question
For a given $n$, does there exist a $2n \times 2n$ matrix with entries in $\{1, 0, -1\}$ having orthogonal ... | There is no such matrix if $n\equiv 3\pmod 4$.
Suppose otherwise. Each column represents a vector of length $\sqrt n$. Since those vectors are pairwise orthogonal, their sum is a vector whose scalar square is $2n^2$.
On the other hand, the sum of all columns has odd entries, so their squares are all congruent to $1$ m... | {
"language": "en",
"url": "https://mathoverflow.net/questions/421691",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 3,
"answer_id": 2
} |
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Fractions in Questions and Answers
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