Unsolved Mathematical Problems


Several prizes are waiting to be awarded for outstanding mathematical achievement. The most prestigious mathematical award is the Fields medal. Other awards are the Wolf Prize from the Wolf Foundation of Israel, the Leroy P. Steele Prize of the American Mathematical Society, the Bôcher Prize, Cole Prizes in algebra and number theory, and the Delbert Ray Fulkerson Prize, all presented by the American Mathematical Society.

institueIn 2000, The Clay Mathematics Institute of Cambridge, Massachusetts (CMI) named seven “Millennium Prize Problems,” selected by focusing on important classic questions in mathematics that have resisted solution over the years. A $7 million prize fund was established for the solution to these problems, with $1 million allocated to each. The problems are:



1. Riemann hypothesis
2. Poincaré conjecture (the correct solution was confirmed in 2006, the prize was turned down)
3. Hodge conjecture
4. Swinnerton-Dyer conjecture
5. solution of the Navier-Stokes equations
6. formulation of Yang-Mills theory
7. determination of whether NP-problems are actually P-problems.

There are too many unsolved mathematical problems to include them all in this single article. Listed below are some of the unsolved mathematical problems by category:


Homological conjectures in commutative algebra
Hilbert’s fifteenth and sixteenth problem
Hadamard conjecture
Jacobson’s conjecture
Zauner’s conjecture
Wild Problem
Köthe conjecture
Birch–Tate conjecture
Serre’s conjecture II
Bombieri–Lang conjecture
Farrell–Jones conjecture
Bost conjecture
Uniformity conjecture
Kaplansky’s conjecture
Kummer–Vandiver conjecture
Serre’s multiplicity conjectures
Pierce–Birkhoff conjecture
Eilenberg–Ganea conjecture
Green’s conjecture
Grothendieck–Katz p-curvature conjecture
Sendov’s conjecture

Algebraic geometry

André–Oort conjecture
Bass conjecture
Deligne conjecture
Fröberg conjecture
Fujita conjecture
Hartshorne conjectures
Manin conjecture
Nakai conjecture
Resolution of singularities in characteristic p
Standard conjectures on algebraic cycles
Section conjecture
Tate conjecture
Virasoro conjecture
Whitehead conjecture
Zariski multiplicity conjecture

Algebraic number theory

Are there infinitely many real quadratic number fields with unique factorization?
Characterize all algebraic number fields that have some power basis.
Stark conjectures (including Brumer–Stark conjecture)


The Jacobian conjecture
Schanuel’s conjecture and four exponentials conjecture
Lehmer’s conjecture
Pompeiu problem
Khabibullin’s conjecture on integral inequalities
Hilbert’s thirteenth problem
Vitushkin’s conjecture

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
The Kobon triangle problem
The McMullen problem
Ulam’s packing conjecture
Filling area conjecture
Hopf conjecture
Kakeya conjecture

Euclidean geometry

The Einstein problem
Inscribed square problem
Moser’s worm problem
The moving sofa problem
Dürer’s conjecture
The Thomson problem
Pentagonal tiling
Falconer’s conjecture
Circle packing in an equilateral triangle
Circle packing in an isosceles right triangle

Dynamical systems

Furstenberg conjecture
Margulis conjecture
MLC conjecture
Weinstein conjecture
Is every reversible cellular automaton in three or more dimensions locally reversible?

Graph theory

Barnette’s conjecture
The Erdős–Gyárfás conjecture
The Erdős–Hajnal conjecture
The Hadwiger conjecture
The Erdős–Faber–Lovász conjecture
Harborth’s conjecture
The total coloring conjecture
Hadwiger conjecture (graph theory)
The list coloring conjecture
The Ringel–Kotzig conjecture
The Hadwiger–Nelson problem
Lovász conjecture
Tutte’s conjectures
Petersen coloring conjecture
The reconstruction conjecture
The cycle double cover conjecture
Does a Moore graph with girth 5 and degree 57 exist?
Conway’s thrackle conjecture
Negami’s conjecture
The Blankenship–Oporowski conjecture
Hedetniemi’s conjecture
Vizing’s conjecture

Group theory

Is every finitely presented periodic group finite?
The inverse Galois problem
Is every group surjunctive?
Andrews–Curtis conjecture
Herzog–Schönheim conjecture
Does generalized moonshine exist?

Model theory

Vaught’s conjecture
The Cherlin–Zilber conjecture
Determine the structure of Keisler’s order
The stable field conjecture
The Stable Forking Conjecture for simple theories
For which number fields does Hilbert’s tenth problem hold?
Shelah’s eventual Categority conjecture
Kueker’s conjecture
Lachlan’s decision problem
Do the Henson graphs have the finite model property?
The universality problem for C-free graphs
The universality spectrum problem

Partial differential equations

Regularity of solutions of Vlasov–Maxwell equations
Regularity of solutions of Euler equations

Ramsey theory

The values of the Ramsey numbers
The values of the Van der Waerden numbers
Erdős–Burr conjecture

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