# Tagged Questions

If it turns out that a problem is equivalent to a known open problem, then the open-problem tag is added. After that, the question essentially becomes, "What is known about this problem? What are some possible ways to approach this problem? What are some ways that people have tried to attack it ...

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### Polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$?

Is there any polynomial $f(x,y)\in{\mathbb Q}[x,y]{}\$ such that $f:\mathbb{Q}\times\mathbb{Q} \rightarrow\mathbb{Q}$ is a bijection?
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### If $2^x$and $3^x$ are integers, must $x$ be as well?

I'm fascinated by this open problem (if it is indeed still that) and every few years I try to check up on its status. Some background: Let $x$ be a positive real number. If $n^x$ is an integer for ...
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### Is it known that the ring of periods is not a field?

I have just learned here that we know numbers that are not periods; is it known meanwhile that the ring of periods is not a field? I know that it is conjectured that $1/\pi$ is not a period, but the ...
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### Volumes of Sets of Constant Width in High Dimensions

Background The n dimensional Euclidean ball of radius 1/2 has width 1 in every direction. Namely, when you consider a pair of parallel tangent hyperplanes in any direction the distance between them ...
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### Open problems with monetary rewards

Since the old days, many mathematicians have been used to attaching monetary rewards to problems they admit are difficult. Their reasons could be to draw other mathematicians' attention, to express ...
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The following question was a research exercise (i.e. an open problem) in R. Graham, D.E. Knuth, and O. Patashnik, "Concrete Mathematics", 1988, chapter 1. It is easy to show that $$\sum_{1 \leq k } (... 8answers 8k views ### Is there a complex structure on the 6-sphere? I don't know who first asked this question, but it's a question that I think many differential and complex geometers have tried to answer because it sounds so simple and fundamental. There are even a ... 3answers 3k views ### Does linearization of categories reflect isomorphism? Given a category C and a commutative ring R, denote by RC the R-linearization: this is the category enriched over R-modules which has the same objects as C, but the morphism module between ... 6answers 11k views ### Is Thompson's Group F amenable? Last year a paper on the arXiv (Akhmedov) claimed that Thompson's group F is not amenable, while another paper, published in the journal "Infinite dimensional analysis, quantum probability, and ... 9answers 6k views ### The “sensitivity” of 2-colorings of the d-dimensional integer lattice Consider the d-dimensional integer lattice, Z^d. Call two points in Z^d "neighbors" if their Euclidean distance is 1 (i.e., if they differ by 1 on exactly one coordinate). Let C be a two-... 2answers 2k views ### vector balancing problem I believe the solution posted to the arXiv on June 17 by Marcus, Spielman, and Srivastava is correct. This problem may be hard, so I don't expect an off-the-cuff solution. But can anyone suggest ... 1answer 2k views ### A function whose fixed points are the primes If a(n) = (\text{largest proper divisor of } n), let f:\mathbb{N} \setminus \{ 0,1\} \to \mathbb{N} be defined by f(n) = n+a(n)-1. For instance, f(100)=100+50-1=149. Clearly the fixed points ... 30answers 6k views ### Fundamental problems whose solution seems completely out of reach [closed] In many areas of mathematics there are fundamental problems that are embarrasingly natural or simple to state, but whose solution seem so out of reach that they are barely mentioned in the literature ... 14answers 7k views ### What are some of the big open problems in 3-manifold theory? From what I understand, the geometrization theorem and its proof helped to settle a lot of outstanding questions about the geometry and topology of 3-manifolds, but there still seems to be quite a lot ... 2answers 4k views ### Polynomials having a common root with their derivatives Here is a question someone asked me a couple of years ago. I remember having spent a day or two thinking about it but did not manage to solve it. This may be an open problem, in which case I'd be ... 2answers 1k views ### Local structure of rational varieties I've been asked this question by a colleague who's not an algebraic geometer; we both feel that the answer should be "no", but I don't have a clue how to prove it. Here's the question: let X be a ... 3answers 2k views ### Shortest closed curve to inspect a sphere Let S be a sphere in \mathbb{R}^3. Let C be a closed curve in \mathbb{R}^3 disjoint from and exterior to S which has the property that every point x on S is visible to some point y of ... 4answers 7k views ### Do there exist chess positions that require exponentially many moves to reach? By "chess" here I mean chess played on an n\times n board with an unbounded number of (non-king) pieces. Some care is needed if you want to generalize some of the subtler rules of chess to an n\... 12answers 2k views ### Can a discrete set of the plane of uniform density intersect all large triangles? Let S be a discrete subset of the Euclidean plane such that the number of points in a large disc is approximately equal to the area of the disc. Does the complement of S necessarily contain triangles ... 0answers 682 views ### Which region in the plane with a given area has the most domino tilings? I just finished teaching a class in combinatorics in which I included a fairly easy upper bound on the number of domino tilings of a region in the plane as a function of its area. So this led to ... 5answers 4k views ### Can N^2 have only digits 0 and 1, other than N=10^k? Pablo Solis asked this at a recent 20 questions seminar at Berkeley. Is there a positive integer N, not of the form 10^k, such that the digits of N^2 are all 0's and 1's? It seems very unlikely,... 8answers 6k views ### Series whose convergence is not known For most of the mathematical concepts I learn, it has more or less always been possible to find (at least google and find) unsolved problems pertaining to that specific concept. Keeping a bag of ... 1answer 2k views ### improving known bounds for Pierce expansions; cash prize Here's a problem that I thought of back in 1978 or so, and only a little progress has been made on it since then. I still think about it from time to time, but probably not that many people have ... 3answers 3k views ### Difficult examples for Frankl's union-closed conjecture Frankl's well-known union-closed conjecture states that if F is a finite family of sets that is closed under taking unions (that is, if A and B belong to the family then so does A\cup B), then there ... 4answers 3k views ### The maximum of a polynomial on the unit circle Encouraged by the progress made in a recently posted MO problem, here is a "conceptually related" problem originating from a 2003 joint paper of Sergei Konyagin and myself. Suppose we are given n ... 5answers 3k views ### Factorials in Pascals Triangle Hi, I asked this question of Keith Conrad, and he suggested that I try posting here. One of my students observed that the only instances of factorials in the interior of Pascal's triangle are \... 0answers 1k views ### Set-theoretic reformulation of the invariant subspace problem The invariant subspace problem (ISP) for Hilbert spaces asks whether every bounded linear operator A on l^2 (with complex scalars) must have a closed invariant subspace other than \{0\} and l^2... 8answers 3k views ### The shortest path in first passage percolation Consider a square planar grid. (The vertices are pair of points in the plane with integer coordinates and two vertices are adjacent if they agree in one coordinate and differ by one in the other.) ... 3answers 1k views ### Is the fixed point property for posets preserved by products? Recall that a partially ordered set (poset) P has the fixed point property (FPP) if any order preserving function f:P\longrightarrow P has a fixed point. Theorem. Suppose P and Q are posets ... 2answers 2k views ### Euler and the Four-Squares Theorem There are several questions in the Euler-Goldbach correspondence that I am unable to answer. Sometimes it does not take very much: in his letter to Goldbach dated June 9th, 1750, Euler conjectured ... 6answers 3k views ### Can we color Z^+ with n colors such that a, 2a, …, na all have different colors for all a? For example for n=2 coloring odd numbers red, numbers of the form 4k+2 blue and so on works. This problem was posed in the KoMaL for n+1 prime, if I know well by Geza Kos. I verified it for all n<... 1answer 1k views ### Density of values of polynomials in two variables This question is a reposting of a comment I made on Polynomial representing all nonnegative integers. Let f(x,y)\in \mathbb Q[x,y] such that f(\mathbb Z\times \mathbb Z) is a subset of \mathbb N... 8answers 6k views ### What are some important but still unsolved problems in mathematical logic? In the past, First order logic and its completeness and whether arithmetic is complete was a major unsolved issues in logic . All of these problems were solved by Godel. Later on, independence of ... 2answers 1k views ### Example of a compact Kähler manifold with non-finitely generated canonical ring? A celebrated recent theorem of Birkar-Cascini-Hacon-McKernan and Siu says that the canonical ring R(X)=\oplus_{m\geq 0}H^0(X,mK_X) of any smooth algebraic variety X over \mathbb{C} is a finitely ... 9answers 9k views ### What are some open problems in algebraic geometry? What are the open big problems in algebraic geometry and vector bundles? More specifically, I would like to know what are interesting problems related to moduli spaces of vector bundles over ... 2answers 1k views ### A group-theoretic perspective on Frankl's union closed problem Here is a group theoretic phrasing of a special case of the union closed conjecture: Question: Given a finite group G, is there an element of prime power order which is contained in at most half ... 1answer 2k views ### Is the set of primes “translation-finite”? The definition in the title probably needs explaining. I should say that the question itself was an idea I had for someone else's undergraduate research project, but we decided early on it would be ... 1answer 2k views ### Is S^2 x S^4 a complex manifold? As observed by Calabi a long time ago, the manifold S^2\times S^4 admits an almost-complex structure (obtained by embedding it in \mathbb{R}^7 and using the octonionic product), which however is ... 0answers 2k views ### A Combinatorial Abstraction for The “Polynomial Hirsch Conjecture” Consider t disjoint families of subsets of {1,2,…,n}, {\cal F}_1,{\cal F_2},\dots {\cal F_t} . Suppose that (*) For every i \lt j \lt k and every R \in {\cal F}_i, and T \in {\cal F}_k, ... 5answers 1k views ### Surfaces filled densely by a geodesic Which smooth, closed surfaces S \subset \mathbb{R}^3 have no single geodesic \gamma that fills S densely? Say a geodesic \gamma "fills S densely" if the closure of the set of points ... 4answers 2k views ### Are most cubic plane curves over the rationals elliptic? %This is a new version of the original question modified in the light of the answers and comments. The word 'most' in the title is ambiguous. The following is one way of making it precise. Question1:... 5answers 2k views ### Is Lebesgue's “universal covering” problem still open? The following problem has been attributed to Lebesgue. Let "set" denote any subset of the Euclidean plane. What is the greatest lower bound of the diameter of any set which contains a subset congruent ... 2answers 3k views ### Projective Plane of Order 12 I asked this question on the new Theoretical Computer Science "overflow" site, and commenters suggested I ask it here. That question is here, and it contains additional links, which I doubt I can ... 4answers 5k views ### Are nontrivial integer solutions known for x^3+y^3+z^3=3? The Diophantine equation$$x^3+y^3+z^3=3 has four easy integer solutions: $(1,1,1)$ and the three permutations of $(4,4,-5)$. Elsenhans and Jahnel wrote in 2007 that these were all the solutions ...
Recall that a function $f\colon X\times X \to \mathbb{R}_{\ge 0}$ is a metric if it satisfies: definiteness: $f(x,y) = 0$ iff $x=y$, symmetry: $f(x,y)=f(y,x)$, and the triangle inequality: \$f(x,y) \...