List of unsolved problems in mathematics

Some problems belong to more than one discipline and are studied using techniques from different areas.

Various mathematicians and organizations have published and promoted lists of unsolved mathematical problems.

In some cases, the lists have been associated with prizes for the discoverers of solutions.

Of the original seven Millennium Prize Problems listed by the Clay Mathematics Institute in 2000, six remain unsolved to date:[6] The seventh problem, the Poincaré conjecture, was solved by Grigori Perelman in 2003.

[14] However, a generalization called the smooth four-dimensional Poincaré conjecture—that is, whether a four-dimensional topological sphere can have two or more inequivalent smooth structures—is unsolved.

The Riemann zeta function , subject of the Riemann hypothesis [ 13 ]
In the Bloch sphere representation of a qubit , a SIC-POVM forms a regular tetrahedron . Zauner conjectured that analogous structures exist in complex Hilbert spaces of all finite dimensions.
The free Burnside group is finite; in its Cayley graph , shown here, each of its 27 elements is represented by a vertex. The question of which other groups are finite remains open.
A detail of the Mandelbrot set . It is not known whether the Mandelbrot set is locally connected or not.
In three dimensions, the kissing number is 12, because 12 non-overlapping unit spheres can be put into contact with a central unit sphere. (Here, the centers of outer spheres form the vertices of a regular icosahedron .) Kissing numbers are only known exactly in dimensions 1, 2, 3, 4, 8 and 24.
An instance of the Erdős–Faber–Lovász conjecture: a graph formed from four cliques of four vertices each, any two of which intersect in a single vertex, can be four-colored.
6 is a perfect number because it is the sum of its proper positive divisors, 1, 2 and 3. It is not known how many perfect numbers there are, nor if any of them is odd.
The area of the blue region converges to the Euler–Mascheroni constant , which may or may not be a rational number.
Goldbach's conjecture states that all even integers greater than 2 can be written as the sum of two primes. Here this is illustrated for the even integers from 4 to 28.
The unknotting problem asks whether there is an efficient algorithm to identify when the shape presented in a knot diagram is actually the unknot .
Ricci flow , here illustrated with a 2D manifold, was the key tool in Grigori Perelman 's solution of the Poincaré conjecture .