# List of unsolved problems in mathematics

## Millennium Prize Problems

Of the seven Millennium Prize Problems set by the Clay Mathematics Institute, six have yet to be solved:

The seventh problem, the Poincaré conjecture, has been solved. The smooth four-dimensional Poincaré conjecture is still unsolved. That is, can a four-dimensional topological sphere have two or more inequivalent smooth structures?

## Other still-unsolved problems

### Discrete geometry

• Solving the Happy Ending problem for arbitrary $n$
• Finding matching upper and lower bounds for K-sets and halving lines
• The Hadwiger conjecture on covering n-dimensional convex bodies with at most 2n smaller copies

### Combinatorics

• Number of Magic squares (sequence A006052 in OEIS)
• Finding a formula for the probability that two elements chosen at random generate the symmetric group $S_n$
• Frankl’s union-closed sets conjecture: for any family of sets closed under sums there exists an element (of the underlying space) belonging to half or more of the sets
• The Lonely runner conjecture: if $k+1$ runners with pairwise distinct speeds run round a track of unit length, will every runner be “lonely” (that is, be more than a distance $1/(k+1)$ from each other runner) at some time?
• Singmaster’s conjecture: is there a finite upper bound on the multiplicities of the entries greater than 1 in Pascal’s triangle?
• The 1/3–2/3 conjecture: does every finite partially ordered set contain two elements x and y such that the probability that x appears before y in a random linear extension is between 1/3 and 2/3?
• Conway’s thrackle conjecture

### Dynamics

• Fürstenberg conjecture – Is every invariant and ergodic measure for the $\times 2,\times 3$ action on the circle either Lebesgue or atomic?
• Margulis conjecture — Measure classification for diagonalizable actions in higher-rank groups