# Questions tagged [open-problems]

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 before, and with what results?"

**460**

**2**answers

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

**305**

**105**answers

### Not especially famous, long-open problems which anyone can understand

**266**

**7**answers

### Polynomial representing all nonnegative integers

**180**

**8**answers

### Two commuting mappings in the disk

**154**

**30**answers

### Proposals for polymath projects

**118**

**4**answers

### If $2^x $and $3^x$ are integers, must $x$ be as well?

**106**

**0**answers

### Why polynomials with coefficients $0,1$ like to have only factors with $0,1$ coefficients?

**103**

**1**answer

### What is the definition of the function T used in Atiyah's attempted proof of the Riemann Hypothesis?

**96**

**13**answers

### What are the big problems in probability theory?

**89**

**26**answers

### Open problems with monetary rewards

**89**

**24**answers

### Noteworthy, but not so famous conjectures resolved recent years

**79**

**0**answers

### Volumes of Sets of Constant Width in High Dimensions

**76**

**11**answers

### Is there a complex structure on the 6-sphere?

**74**

**0**answers

### Converse to Euclid's fifth postulate

**73**

**6**answers

### Can we cover the unit square by these rectangles?

**70**

**2**answers

### Is it known that the ring of periods is not a field?

**68**

**16**answers

### Important open problems that have already been reduced to a finite but infeasible amount of computation

**67**

**1**answer

### Nontrivial finite group with trivial group homologies?

**65**

**0**answers

### 2, 3, and 4 (a possible fixed point result ?)

**64**

**3**answers

### Does linearization of categories reflect isomorphism?

**59**

**7**answers

### Is Thompson's Group F amenable?

**58**

**1**answer

### Which region in the plane with a given area has the most domino tilings?

**57**

**14**answers

### What are some of the big open problems in 3-manifold theory?

**54**

**1**answer

### Intersecting Family of Triangulations

**52**

**11**answers

### What are some open problems in algebraic geometry?

**52**

**10**answers

### The “sensitivity” of 2-colorings of the d-dimensional integer lattice

**51**

**8**answers

### Series whose convergence is not known

**51**

**9**answers

### What are some important but still unsolved problems in mathematical logic?

**50**

**30**answers

### Fundamental problems whose solution seems completely out of reach [closed]

**50**

**14**answers

### Open problems in Euclidean geometry?

**49**

**2**answers

### vector balancing problem

**48**

**2**answers

### Polynomials having a common root with their derivatives

**48**

**4**answers

### Do there exist chess positions that require exponentially many moves to reach?

**47**

**1**answer

### A function whose fixed points are the primes

**46**

**2**answers

### Local structure of rational varieties

**45**

**13**answers

### PhD dissertations that solve an established open problem

**45**

**0**answers

### Set-theoretic reformulation of the invariant subspace problem

**44**

**6**answers

### Can we color Z^+ with n colors such that a, 2a, …, na all have different colors for all a?

**44**

**5**answers

### Can $N^2$ have only digits 0 and 1, other than $N=10^k$?

**44**

**3**answers

### Shortest closed curve to inspect a sphere

**44**

**4**answers

### Difficult examples for Frankl's union-closed conjecture

**44**

**1**answer

### improving known bounds for Pierce expansions; cash prize

**43**

**5**answers

### Is Lebesgue's “universal covering” problem still open?

**43**

**4**answers

### The maximum of a polynomial on the unit circle

**40**

**12**answers

### Can a discrete set of the plane of uniform density intersect all large triangles?

**40**

**2**answers

### Open problems/questions in representation theory and around?

**37**

**5**answers

### Surfaces filled densely by a geodesic

**37**

**4**answers

### Does pi contain 1000 consecutive zeroes (in base 10)?

**37**

**3**answers

### Is the fixed point property for posets preserved by products?

**37**

**9**answers