Unsolved problems in mathematics

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This article lists some unsolved problems in mathematics. See individual articles for details and sources.

[edit] Millennium Prize Problems

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

Only 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?

[edit] Other still-unsolved problems

[edit] Additive number theory

Goldbach's conjecture and its weak version
The values of $g (k)$ and $G (k)$ in Waring's problem
Collatz conjecture ( $3 n + 1$ conjecture)
Gilbreath's conjecture

[edit] Number theory: prime numbers

Catalan's Mersenne conjecture
Twin prime conjecture
Are there infinitely many prime quadruplets?
Are there infinitely many Mersenne primes (Lenstra-Pomerance-Wagstaff conjecture); equivalently, infinitely many even perfect numbers?
Are there infinitely many Sophie Germain primes?
Are there infinitely many regular primes, is their relative density $e^{-^1\!/_2}$ ?
Are there infinitely many Cullen primes?
Are there infinitely many palindromic primes in base 10?
Are there infinitely many Fibonacci primes?
Is every Fermat number 2^2ⁿ+1 composite for $n > 4$ ?
Is 78,557 the lowest Sierpinski number?
Is 509,203 the lowest Riesel number?
Fortune's conjecture (that no Fortunate number is composite)
Polignac's conjecture
Landau's problems

[edit] General number theory

abc conjecture
Do any odd perfect numbers exist?
Do quasiperfect numbers exist?
Do any odd weird numbers exist?
Do any Lychrel numbers exist?
Is 10 a solitary number?
Do any Taxicab(5, 2, n) exist for n>1?
Brocard's problem: existence of integers, n,m, such that n!+1=m² other than n=4,5,7

[edit] Algebraic number theory

Existence of quadratic number fields being Euclidean but not norm-Euclidean.
Are there an infinite number of real quadratic number fields with unique factorization?

[edit] Discrete geometry

Solving the Happy Ending problem for arbitrary $n$
Finding matching upper and lower bounds for K-sets and halving lines
The Hirsch conjecture on the lengths of shortest paths in the vertices and edges of a convex polytope
The Hadwiger conjecture on covering n-dimensional convex bodies with at most 2ⁿ smaller copies

[edit] Ramsey theory

The values of the Ramsey numbers, particularly $R (5,5)$
The values of the Van der Waerden numbers

[edit] General algebra

[edit] 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 $k$ 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 a finite upper bound on the multiplicities of the entries greater than 1 in Pascal's triangle?

[edit] Graph theory

The Erdős-Gyárfás conjecture on cycles with power-of-two lengths in cubic graphs
The Hadwiger conjecture relating coloring to clique minors
The Erdős–Faber–Lovász conjecture on coloring unions of cliques
The total coloring conjecture
The list coloring conjecture
The Ringel-Kotzig conjecture on graceful labeling of trees
The Hadwiger–Nelson problem on the chromatic number of unit distance graphs
Deriving a closed-form expression for the percolation threshold values, especially $p c$ (square site)
Tutte's conjectures that every bridgeless graph has a nowhere-zero 5-flow and every bridgeless graph without the Petersen graph as a minor has a nowhere-zero 4-flow
The Reconstruction conjecture and New digraph reconstruction conjecture concerning whether or not a graph is recognizable by the vertex deleted subgraphs.
The cycle double cover conjecture that every bridgeless graph has a family of cycles that includes each edge twice.

[edit] Analysis

the Jacobian conjecture
Schanuel's conjecture
Lehmer's conjecture
Pompeiu problem
Is $γ$ (the Euler-Mascheroni constant) irrational?

[edit] Group theory

Is every finitely presented periodic group finite?
The inverse Galois problem
For which positive integers m, n is the free Burnside group B(m,n) finite? In particular, is B(2, 5) finite?

[edit] Set theory

The problem of finding the ultimate core model, one that contains all large cardinals.
If ℵ_ω is a strong limit cardinal, then 2^ℵ_ω < ℵ_ω₁ (see Singular cardinals hypothesis). The best bound, ℵ_ω₄, was obtained by Shelah using his pcf theory.
Woodin's Ω-hypothesis.
Does the consistency of the existence of a strongly compact cardinal imply the consistent existence of a supercompact cardinal?
(Woodin) Does the Generalized Continuum Hypothesis below a strongly compact cardinal imply the Generalized Continuum Hypothesis everywhere?
Does there exist a Jonsson algebra on ℵ_ω?

[edit] Other

Generalized star height problem
Invariant subspace problem
Modeling black hole mergers
Problems in Latin squares
Problems in loop theory and quasigroup theory

[edit] Problems solved recently

Road coloring conjecture (Avraham Trahtman, 2007)
The Angel problem (Various independent proofs, 2006)
Stanley–Wilf conjecture (Gábor Tardos and Adam Marcus, 2004)
Green–Tao theorem (Ben J. Green and Terence Tao, 2004)
Poincaré conjecture (Solution by Grigori Perelman in 2002 now confirmed)
Catalan's conjecture (Preda Mihăilescu, 2002)
Kato's conjecture (Auscher, Hofmann, Lacey, McIntosh, and Tchamitchian, 2001)
The Langlands program for function fields (Laurent Lafforgue, 1999)
Taniyama-Shimura conjecture (Wiles, Breuil, Conrad, Diamond, and Taylor, 1999)
Kepler conjecture (Thomas Hales, 1998)
Milnor conjecture (Vladimir Voevodsky, 1996)
Fermat's Last Theorem (Andrew Wiles, 1994)
Bieberbach conjecture (Louis de Branges, 1985)
Four color theorem (Appel and Haken, 1977)

[edit] See also

[edit] References

[edit] Books discussing unsolved problems

Fan Chung; Ron Graham (1999). Erdos on Graphs: His Legacy of Unsolved Problems. AK Peters. ISBN 1-56881-111-X.
Hallard T. Croft; Kenneth J. Falconer; Richard K. Guy (1994). Unsolved Problems in Geometry. Springer. ISBN 0-387-97506-3.
Richard K. Guy (2004). Unsolved Problems in Number Theory. Springer. ISBN 0-387-20860-7.
Victor Klee; Stan Wagon (1996). Old and New Unsolved Problems in Plane Geometry and Number Theory. The Mathematical Association of America. ISBN 0-88385-315-9.
Marcus Du Sautoy (2003). The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics. Harper Collins. ISBN 0060935588.
Keith Devlin (2006). The Millennium Problems - The Seven Greatest Unsolved* Mathematical Puzzles Of Our Time. Barnes & Noble. ISBN 0-7607-8659-8.

[edit] Books discussing recently solved problems

Simon Singh (2002). Fermat's Last Theorem. Fourth Estate. ISBN 1841157910.
Donal O'Shea (2007). The Poincaré Conjecture. Penguin. ISBN 978-1-846-14012-9.
George G. Szpiro (2003). Kepler's Conjecture. Wiley. ISBN 0-471-08601-0.
Mark Ronan (2006). Symmetry and the Monster. Oxford. ISBN 0-19-280722-6.

[edit] Resources

Open Problem Garden The collection of open problems in mathematics build on the principle of user editable ("wiki") site