Unsolved problems in mathematics

This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please improve this article by introducing more precise citations where appropriate. (May 2008)

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

Contents 1 Millennium Prize Problems 2 Other still-unsolved problems 2.1 Additive number theory 2.2 Number theory: prime numbers 2.3 General number theory 2.4 Algebraic number theory 2.5 Discrete geometry 2.6 Ramsey theory 2.7 General algebra 2.8 Combinatorics 2.9 Graph theory 2.10 Analysis 2.11 Group theory 2.12 Set theory 2.13 Other 3 Problems solved recently 4 See also 5 References 6 Books discussing unsolved problems 7 Books discussing recently solved problems 8 Resources

Millennium Prize Problems

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

• P versus NP
• The Hodge conjecture
• The Riemann hypothesis
• Yang–Mills existence and mass gap
• Navier-Stokes existence and smoothness
• The Birch and Swinnerton-Dyer conjecture

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?

Other still-unsolved problems

Additive number theory

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

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 ?
• 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 2²ⁿ+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

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
• Distribution and upper bound of mimic numbers

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?

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

Ramsey theory

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

General algebra

• Hilbert's sixteenth problem
• Hadamard conjecture
• existence of perfect cuboids

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?

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.

Analysis

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

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?

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?

Other

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

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)

See also

• Hilbert's 23 problems
• Smale's problems
• Timeline of mathematics

References

• Unsolved Problems in Number Theory, Logic and Cryptography
• Clay Institute Millennium Prize
• Unsolved problems page at MathWorld
• Winkelmann, Jörg, "Some Mathematical Problems". 9 March 2006.
• List of links to unsolved problems in mathematics, prizes and research.

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. 

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. 

Resources

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

v • d • e Unsolved scientific problems astronomy  • chemistry  • computer science  • economics  • linguistics  • mathematics  • neuroscience  • philosophy  • physics  • statistics