Questions tagged [counterexamples]

A counterexample is an example that disproves a mathematical conjecture or a purported theorem. For example, the Peterson graph is a counterexample to many seemingly plausible conjectures in Graph Theory.

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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+\...
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Conjectured primality test for specific class of $N=k \cdot 6^n+1$

Can you provide a proof or a counterexample for the claim given below? Inspired by Theorem 5 in this paper I have formulated the following claim: Let $N=k \cdot 6^n+1$ , $k<6^n$ and $\...
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114 views

Recursively obtained hard Diophantine equation for “Baseless numbers”

An equivalent problem was originally asked on MSE as Does every number base have at least one “Baseless number”?, but did not receive any answers that would help answer the main question about "...
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Convergence in $C_c$ but not in $C$

Let $C_c(\mathbb{R})$ be the set of compactly-supported continuous functions on $\mathbb{R}$. We can view this with a number of different topologies but I have my eye on two in particular. Let $X$ ...
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On which regions can Green's theorem not be applied?

In elementary calculus texts, Green's theorem is proved for regions enclosed by piecewise smooth, simple closed curves (and by extension, finite unions of such regions), including regions that are not ...
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Is there a finite extension with a non-trivial class group of any PID?

Let $R$ be a PID with infinitely many prime ideals. Does there always exist a finite extension $R\subset R'$ with $R'$ being a Dedekind domain with a non-trivial class group?
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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}\...
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2answers
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Counterexample for absolute summability of autocovariances of strictly stationary strongly mixing sequence

Suppose $(X_i)_{i\in\mathbb{Z}}$ is a strictly stationary, strongly (i.e. $\alpha-$)mixing sequence of real random variables. If we have $\mathbb{E}[|X_1|^{2+\epsilon}]<\infty$ for some $\epsilon&...
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1answer
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Is there an i.i.d sequence in the unit cube $[-1,1]^d$ with $\mathbb E \left[ \Big \| \sum_{i=1}^N X_N \Big \|_\infty\right] = \sqrt {dN}$?

There are loads of concentration results for sums of scalar-valued independent sums $X_1,X_2,\ldots, X_N$ with $\mathbb E[X_n]=0$. For example Hoeffding's Inequality says if all $|X_1|\le 1$ then $\...
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139 views

Non-existent matrices with “essential zeros”

Is there a non-constant continous function $f:\mathbb{R}\rightarrow \mathbb{R}$ and matrices $A=\begin{pmatrix} a_1 & 0\\ 0 & a_2\\ \end{pmatrix}$ and $B=\begin{pmatrix} b_1 & 0\\ 0 & ...
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315 views

Conjectured primality test for specific class of $N=4kp^n+1$

Can you provide a proof or counterexample for the following claim? Let $P_m(x)=2^{-m}\cdot((x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4})^m)$ . Let $N= 4kp^{n}+1 $ where $k$ is a positive natural number , $...
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1answer
166 views

Most general form of Jensen's inequality

What is the most general form of Jensen's inequality? Wikipedia gives for example this more general form, which holds in every topological vector space. Are there even more general forms, for ...
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716 views

Can you give an example of two projective morphisms of schemes whose composition is not projective?

Grothendieck and Dieudonné prove in $EGA_{II}$ (Proposition 5.5.5.(ii), page 105) that if $f:X\to Y, g:Y\to Z$ are projective morphisms of schemes and if $Z$ is separated and quasi-compact, or if ...
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391 views

Are epimorphic endomorphisms of noetherian commutative rings always injective?

This question was asked, but not answered, on Mathematics Stackexchange. [In this post "ring" means "commutative ring with one".] Let $A$ be a noetherian ring, and let $f:A\to A$ be an endomorphism ...
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Can different knots have the same numbers of quandle colorings for all quandles?

Let $K_1$ and $K_2$ be two knots such that for all finite quandles $X$, the number of colorings of $K_1$ by $X$ is the same as the number of colorings of $K_2$ by $X$. Then my question is, must $K_1$ ...
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A linear map satisfying the given property

Let $A$ and $B$ be two Banach algebras such that $B$ is a Banach $A$-bimodue and $T:A\rightarrow B$ a linear map satisfying $T(aa')=aT(a')+T(a)a'+T(a)T(a')$ for all $a,a'\in A$. If the algerba ...
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1answer
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External tensor product of irreducible representations is not irreducible?

I'm writing up some notes, and I realize I don't have a counterexample for something I suspect is false. Dubious claim: If $(\pi, V)$ and $(\rho, W)$ are irreducible representations of two groups $G$...
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Examples for a Golomb's result, and rationals as $\sum_{n\geq 1}\frac{|G_n|}{P(n)}$, where $G_n$ are Gregory coefficients and $P(x)$ a polynomial

After I was stuying the first pages of a chapter of the book [1], in particular the statement of Corollary 10.3 and its proof, I wondered what can be interesting examples of irrational numbers that ...
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Equations involving quasiperfect numbers: a first search of odd solutions for this type of equations or well succinct reasonings about these

In this post we study the following equations that involve quasiperfect numbers, denoted as $x$, that are integers such that the sum of all its positive divisors is equals to $2x+1$, and certain ...
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1answer
137 views

Does $\sum_ia\cap b_i=a\cap(\sum_ib_i)$ and $a(\bigcap_i b_i)=\bigcap_iab_i$ for infinite sums and intersections in arithmetic rings (Prufer domains)?

Note: Please let me know if this question is too basic for MathOverflow. It is about a subject commonly taught in graduate school (commutative algebra), and is based in large part on a (very ...
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Multiplication and division by a morphism under the “inner composition” in closed monoidal categories

I asked this a week ago at math.stackexchange, without success, so I hope it will be appropriate here. Let ${\mathcal C}$ be a symmetric closed monoidal category, and let me denote the internal hom-...
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2answers
248 views

Periodic orbit for certain vector field on $S^3$ (à la Seifert conjecture)

The standard frame for $S^3$ consists of $X_i,X_j,X_k$ with $X_i(a)=ia, X_j(a)=ja, X_k(a)=ka$ where $i,j,k$ are standard quaternion numbers, $a\in S^3$, and the multiplication is the quaternion ...
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179 views

Conjectured primality test for numbers of the form $N=4 \cdot 3^n-1$

This is a repost of this question. Can you provide proof or counterexample for the claim given below? Inspired by Lucas-Lehmer primality test I have formulated the following claim: Let $P_m(x)=2^{...
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151 views

Examples of automorphic representations to keep in mind

I have recently started studying the automorphic science and find it somewhat hard to form intuition. Can we have a list of examples of automorphic representations that you usually use to test a new ...
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1answer
383 views

pair of injective morphisms of simplicial groups

Let $X$ and $Y$ be two simplicial groups such that there exists $f:X\rightarrow Y$ and $g:Y\rightarrow X $ two injective morphisms of simplicial groups. Suppose that $X$ is contractible, does it ...
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1answer
87 views

Is there a locally countable and weakly Lindelöf space which is not ccc

Is there a locally countable and weakly Lindelöf space which is not ccc? A space $X$ is locally countable if for each point $x\in X$ there is an open neighbourhood $O_x$ of $x$ such that $|O_x| < \...
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111 views

A complex limit cycle not intersecting the real plane(2)

Inspired by this question and the counter example provided in its answer we ask: Is there a polynomial vector field on $\mathbb{R}^2$ such that after complexification of the equation, the ...
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1answer
110 views

Is the continuous dual of a topological chain complex chain equivalent to the algebraic dual?

I apologize in advance if this is a naive question. Def: A topological chain complex is a chain complex of topological $\mathbb{R}$-vector spaces such that the boundary maps are continuous. Let $C$ ...
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Ricci flow on locally symmetric noncompact manifold

As it is mentioned by Deane Yang in Ricci flow preserves locally symmetry along the flow, we know the local symmetry is preserved under the Ricci flow on the compact manifold since we have the ...
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3answers
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Intuition behind counterexample of Euler's sum of powers conjecture

I was stunned when I first saw the article Counterexample to Euler's conjecture on sums of like powers by L. J. Lander and T. R. Parkin:. How was it possible in 1966 to go through the sheer ...
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1answer
111 views

A special oscillatory orbit in space

Edit: According to the comment of Prof. Eremenko I revise the question. 19 years ago, I have heard the following problem from a specialist of dynamical system. During these 19 years, I was in contact ...
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1answer
124 views

Banach algebra $A$ without an approximate identity but $A^2=A$

Please help me with the following question. What are some examples of Banach algebra $A$ satisfying the following two conditions? $1$.$ A $ does not have an approximate identity. $2$. $A^2=A$. ...
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1answer
60 views

Is the Scott topology generated by the ideals as the closed sets?

Let $X$ be a directed-complete partial order, or even a complete lattice. A subset $S\subseteq X$ is called Scott-closed if and only if it is: Downward-closed: $y\in S$ and $x\le y$ implies $x\in S$; ...
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1answer
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Is the projection onto the regular image an epimorphism?

Let $f:X\to Y$ be a morphism in a category $\mathcal{C}$. Let $m:I\hookrightarrow Y$ be the regular image of $f$. This means that $f$ can be written as $f=m\circ e$, with $m$ regular mono (i.e. being ...
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An example of a Banach algebra with a specified property

I asked this question (https://math.stackexchange.com/questions/3076735/an-example-of-a-banach-algebra-satisfying-given-conditions) but unfortunately no one answered it. Please help me to find an ...
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Counterexample in Kolmogorov theorem about existence of almost surely continuous modification

I want to understand this Kolmogorov theorem about existence of almost surely continuous modification: A process $\{\xi_t, \in[0,T]\}$ admits an almost surely continuous modification if there exist ...
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Is a scheme Noetherian if its topological space and its stalks are?

Is a scheme being Noetherian equivalent to the underlying topological space being Noetherian and all its stalks being Noetherian?
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1answer
847 views

If $\text{dim}(X \times X) = 2\text{dim}(X)$, does $\text{dim}(X^n) = n\text{dim}(X)$?

I have been learning some (topological) dimension theory and have gotten through most of the basic material, at this point, and am about to start looking at papers. In particular, I want to get ...
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1answer
652 views

Intersection of nested open ball in complete metric spaces is nonempty?

My question is that whether the following statement is true or not. In a complete metric space $(X, d)$, if a sequence of open balls $\{B(x_i, r_i)\}_{i=1}^\infty$ satisfies $$ \exists \epsilon > ...
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1answer
640 views

Primality test for generalized Fermat numbers

This question is successor of Primality test for specific class of generalized Fermat numbers . Can you provide a proof or a counterexample for the claim given below? Inspired by Lucas–Lehmer–Riesel ...
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1answer
151 views

Does regular $G_\delta$ imply normal?

I'm trying to prove that if every closed set in a topological space is regular $G_\delta$, then the space is normal. By regular $G_\delta$, I mean for any closed set $A$, there exists a countable ...
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1answer
314 views

Primality test for specific class of $N=k \cdot b^n-1$

This question is successor of Compositeness test for specific class of $N=k \cdot b^n-1$ . Can you provide a proof or a counterexample to the following claim : Let $P_m(x)=2^{-m}\cdot \left(\left(...
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2answers
651 views

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 $N=k\cdot 2^n+1$...
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0answers
14 views

Reducing the Dimension of Rectangular Assignment Matrices

Question: Given a weighted complete bipartite graph $K_{m,n}(U,V,E)$, $m^2\lt n$, are there any counter examples to the assumption, that the edges of the minimum weight maximal bipartite ...
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7answers
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Insights from disproofs after counterexamples have been given

Some conjectures are disproved by a single counter-example and garner little or no further interest or study, such as (to my knowledge) Euler's conjecture in number theory that at least $n$ $n^{th}$ ...
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2answers
600 views

Example of a ring with non-finitely generated unit group?

The well known Dirichlet's unit theorem states that the unit group of a maximal order in a quadratic number field is finitely generated of rank blah blah blah. I think it's pretty naive to expect a ...
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0answers
266 views

Conjectured initial values of Inkeri's primality test for Fermat numbers

This is a repost of this question . Can you provide a proof or a counterexample to the claim given below ? First , we shall give a definition of the Inkeri's primality test for Fermat numbers : ...
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1answer
106 views

Average edge-cost optimality of minimum spanning trees

Are there counter examples to the conjecture, that in a complete, finite and symmetric weighted graph $G\left(V,E,\omega\right),\ E=\lbrace \lbrace i,j\rbrace\subset V\times V\rbrace,\ \omega:E\ni e\...
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2answers
328 views

A locally presentable locally cartesian closed category that is not a quasitopos

This question asks for a locally presentable locally cartesian closed category that is not a topos. All the answers given (at least in the 1-categorical case) are quasitoposes. What is an example of ...
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1answer
268 views

Example of a locally presentable locally cartesian closed category which is not a topos?

The only way I know to get a locally cartesian closed category which is not a topos is to start with a topos and then throw out some objects so that the category is not sufficiently cocomplete to be a ...