List of unsolved problems in mathematics

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

Millennium Prize Problems[edit]

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

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

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[edit]

Additive number theory[edit]

• Beal's conjecture
• Fermat–Catalan conjecture
• Goldbach's conjecture (Proof claimed for weak version in 2013)
• The values of g(k) and G(k) in Waring's problem
• Collatz conjecture (3n + 1 conjecture)
• Lander, Parkin, and Selfridge conjecture
• Diophantine quintuples
• Gilbreath's conjecture
• Erdős conjecture on arithmetic progressions
• Erdős–Turán conjecture on additive bases
• Pollock octahedral numbers conjecture

Algebra[edit]

• Hilbert's sixteenth problem
• Hadamard conjecture
• Existence of perfect cuboids
• Zauner's conjecture: existence of SIC-POVMs in all dimensions

Algebraic geometry[edit]

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

Algebraic number theory[edit]

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

Analysis[edit]

• The Jacobian conjecture
• Schanuel's conjecture
• Lehmer's conjecture
• Pompeiu problem
• Are  (the Euler–Mascheroni constant), π + e, π − e, πe, π/e, π^e, π^√2, π^π, e^π2, ln π, 2^e, e^e, Catalan's constant or Khinchin's constant rational, algebraic irrational, ortranscendental? What is the irrationality measure of each of these numbers?^{[1][2][3][4][5][6][7][8]}
• The Khabibullin’s conjecture on integral inequalities

Combinatorics[edit]

• Number of magic squares (sequence A006052 in OEIS)
• Finding a formula for the probability that two elements chosen at random generate the symmetric group 
• 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  runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be at least a distance  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 that is not totally ordered contain two elements x and y such that the probability that x appears before yin a random linear extension is between 1/3 and 2/3?

Discrete geometry[edit]

• Solving the Happy Ending problem for arbitrary 
• 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
• The Kobon triangle problem on triangles in line arrangements
• The McMullen problem on projectively transforming sets of points into convex position
• Ulam's packing conjecture about the identity of the worst-packing convex solid

Euclidean geometry[edit]

• The einstein problem – does there exist a two-dimensional shape that forms the prototile for an aperiodic tiling, but not for any periodic tiling?^[9]
• Inscribed square problem – does every Jordan curve have an inscribed square?^[10]
• Moser's worm problem – what is the smallest area of a shape that can cover every unit-length curve in the plane?^[11]
• The moving sofa problem – what is the largest area of a shape that can be maneuvered through a unit-width L-shaped corridor?^[12]
• Shephard's conjecture – does every convex polyhedron have a net?^[13]

Dynamical systems[edit]

• Furstenberg conjecture – Is every invariant and ergodic measure for the  action on the circle either Lebesgue or atomic?
• Margulis conjecture — Measure classification for diagonalizable actions in higher-rank groups
• MLC conjecture – Is the Mandelbrot set locally connected ?

Graph theory[edit]

• Barnette's conjecture that every cubic bipartite three-connected planar graph has a Hamiltonian cycle
• The Erdős–Gyárfás conjecture on cycles with power-of-two lengths in cubic graphs
• The Erdős–Hajnal conjecture on finding large homogeneous sets in graphs with a forbidden induced subgraph
• 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  (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.
• Does a Moore graph with girth 5 and degree 57 exist?
• Conway's thrackle conjecture
• Negami's conjecture on the characterization of graphs with planar covers
• The Blankenship–Oporowski conjecture on the book thickness of subdivisions

Group theory[edit]

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

Model theory[edit]

• Vaught's conjecture
• The Cherlin–Zilber conjecture: A simple group whose first-order theory is stable in  is a simple algebraic group over an algebraically closed field.
• The Main Gap conjecture, e.g. for uncountable first order theories, for AECs, and for -saturated models of a countable theory.^[14]
• Determine the structure of Keisler's order^[15][16]
• The stable field conjecture: every infinite field with a stable first-order theory is separably closed.
• Is the theory of the field of Laurent series over  decidable? of the field of polynomials over ?
• (BMTO) Is the Borel monadic theory of the real order decidable? (MTWO) Is the monadic theory of well-ordering consistently decidable?^[17]
• The Stable Forking Conjecture for simple theories^[18]
• For which number fields does Hilbert's tenth problem hold?
• Assume K is the class of models of a countable first order theory omitting countably many types. If K has a model of cardinality  does it have a model of cardinality continuum?^[19]
• Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?^[20]
• If the class of atomic models of a complete first order theory is categorical in the , is it categorical in every cardinal?^[21][22]
• Is every infinite, minimal field of characteristic zero algebraically closed? (minimal = no proper elementary substructure)
• Kueker's conjecture^[23]
• Does there exist an o-minimal first order theory with a trans-exponential (rapid growth) function?
• Lachlan's decision problem
• Does a finitely presented homogeneous structure for a finite relational language have finitely many reducts?
• Do the Henson graphs have the finite model property? (e.g. triangle-free graphs)
• The universality problem for C-free graphs: For which finite sets C of graphs does the class of C-free countable graphs have a universal member under strong embeddings?^[24]
• The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?^[25]

Number theory (general)[edit]

• abc conjecture (Proof claimed in 2012, currently under review.)
• Carmichael's totient function conjecture
• Erdős–Straus conjecture
• Do any odd perfect numbers exist?
• Are there infinitely many perfect numbers?
• 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
• Littlewood conjecture
• Congruent number problem (a corollary to Birch and Swinnerton-Dyer conjecture, per Tunnell's theorem)
• Lehmer's totient problem: if φ(n) divides n − 1, must n be prime?
• Are there infinitely many amicable numbers?
• Are there any pairs of relatively prime amicable numbers?
• The Gauss circle problem – how far can the number of integer points in a circle centered at the origin be from the area of the circle?

Number theory (prime numbers)[edit]

• Catalan's Mersenne conjecture
• Twin prime conjecture
• The Gaussian moat problem: is it possible to find an infinite sequence of distinct Gaussian prime numbers such that the difference between consecutive numbers in the sequence is bounded?
• 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 Wagstaff primes?
• Are there infinitely many Sophie Germain primes?
• Are there infinitely many regular primes, and if so is their relative density ?
• Are there infinitely many Cullen primes?
• Are there infinitely many Woodall primes?
• Are there infinitely many palindromic primes in base 10?
• Are there infinitely many Fibonacci primes?
• Are all Mersenne numbers of prime index square-free?
• Are there infinitely many Wieferich primes?
• Are there for every a ≥ 2 infinitely many primes p such that a^{p − 1} ≡ 1 (mod p²)?^[26]
• Can a prime p satisfy 2^p − 1 ≡ 1 (mod p²) and 3^p − 1 ≡ 1 (mod p²) simultaneously?^[27]
• Are there infinitely many Wilson primes?
• Are there infinitely many Wolstenholme primes?
• Are there any Wall–Sun–Sun primes?
• Is every Fermat number 2²ⁿ + 1 composite for ?
• Are all Fermat numbers square-free?
• Is 78,557 the lowest Sierpiński number?
• Is 509,203 the lowest Riesel number?
• Fortune's conjecture (that no Fortunate number is composite)
• Polignac's conjecture
• Landau's problems
• Does every prime number appear in the Euclid–Mullin sequence?
• Does the converse of Wolstenholme's theorem hold for all natural numbers?
• Elliott–Halberstam conjecture

Partial differential equations[edit]

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

Ramsey theory[edit]

• The values of the Ramsey numbers, particularly 
• The values of the Van der Waerden numbers

Set theory[edit]

• 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?
• Without assuming the axiom of choice, can a nontrivial elementary embedding V→V exist?
• Does the Generalized Continuum Hypothesis entail  for every singular cardinal ?

Other[edit]

• Invariant subspace problem
• Problems in Latin squares
• Problems in loop theory and quasigroup theory
• Dixmier conjecture
• Baum–Connes conjecture
• Generalized star height problem
• Assorted sphere packing problems, e.g. the densest irregular hypersphere packings
• Closed curve problem: Find (explicit) necessary and sufficient conditions that determine when, given two periodic functions with the same period, the integral curve is closed.^[28]
• Toeplitz' conjecture (open since 1911)

See also: List of conjectures

Problems solved recently[edit]

• Kadison–Singer problem (Adam Marcus, Daniel Spielman and Nikhil Srivastava, 2013)
• Gromov's problem on distortion of knots (John Pardon, 2011)
• Circular law (Terence Tao and Van H. Vu, 2010)
• Hirsch conjecture (Francisco Santos Leal, 2010^[29])
• Serre's modularity conjecture (Chandrashekhar Khare and Jean-Pierre Wintenberger, 2008^[30])
• Heterogeneous tiling conjecture (squaring the plane) (Frederick V. Henle and James M. Henle, 2007)
• Road coloring conjecture (Avraham Trahtman, 2007)
• The Angel problem (Various independent proofs, 2006)
• The Langlands–Shelstad fundamental lemma (Ngô Bảo Châu and Gérard Laumon, 2004)
• Stanley–Wilf conjecture (Gábor Tardos and Adam Marcus, 2004)
• Green–Tao theorem (Ben J. Green and Terence Tao, 2004)
• Cameron–Erdős conjecture (Ben J. Green, 2003, Alexander Sapozhenko, 2003, conjectured by Paul Erdős)^[31]
• Strong perfect graph conjecture (Maria Chudnovsky, Neil Robertson, Paul Seymour and Robin Thomas, 2002)
• Poincaré conjecture (Grigori Perelman, 2002)
• Catalan's conjecture (Preda Mihăilescu, 2002)
• Kato's conjecture (Auscher, Hofmann, Lacey, McIntosh, and Tchamitchian, 2001)
• The Langlands correspondence 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 and Richard Taylor, 1995)
• Bieberbach conjecture (Louis de Branges, 1985)
• Princess and monster game (Shmuel Gal, 1979)
• Four color theorem (Appel and Haken, 1977)

See also[edit]

• Hilbert's problems
• List of conjectures
• List of statements undecidable in ZFC
• Millennium prize problems
• Smale's problems
• Timeline of mathematics

References[edit]

1. Jump up^ Weisstein, Eric W., "Pi", MathWorld.
2. Jump up^ Weisstein, Eric W., "e", MathWorld.
3. Jump up^ Weisstein, Eric W., "Khinchin's Constant", MathWorld.
4. Jump up^ Weisstein, Eric W., "Irrational Number", MathWorld.
5. Jump up^ Weisstein, Eric W., "Transcendental Number", MathWorld.
6. Jump up^ Weisstein, Eric W., "Irrationality Measure", MathWorld.
7. Jump up^ An introduction to irrationality and transcendence methods
8. Jump up^ Some unsolved problems in number theory
9. Jump up^ Socolar, Joshua E. S.; Taylor, Joan M. (2012), "Forcing nonperiodicity with a single tile", The Mathematical Intelligencer 34 (1): 18–28, arXiv:1009.1419, doi:10.1007/s00283-011-9255-y, MR 2902144.
10. Jump up^ Matschke, Benjamin (2014), "A survey on the square peg problem", Notices of the American Mathematical Society 61 (4): 346–253, doi:10.1090/noti1100.
11. Jump up^ Norwood, Rick; Poole, George; Laidacker, Michael (1992), "The worm problem of Leo Moser", Discrete and Computational Geometry 7 (2): 153–162, doi:10.1007/BF02187832,MR 1139077.
12. Jump up^ Wagner, Neal R. (1976), "The Sofa Problem", The American Mathematical Monthly 83 (3): 188–189, doi:10.2307/2977022, JSTOR 2977022
13. Jump up^ Demaine, Erik D.; O'Rourke, Joseph (2007), "Chapter 22. Edge Unfolding of Polyhedra", Geometric Folding Algorithms: Linkages, Origami, Polyhedra, Cambridge University Press, pp. 306–338.
14. Jump up^ Shelah S, Classification Theory, North-Holland, 1990
15. Jump up^ Keisler, HJ, “Ultraproducts which are not saturated.” J. Symb Logic 32 (1967) 23—46.
16. Jump up^ Malliaris M, Shelah S, "A dividing line in simple unstable theories." http://arxiv.org/abs/1208.2140
17. Jump up^ Gurevich, Yuri, "Monadic Second-Order Theories," in J. Barwise, S. Feferman, eds., Model-Theoretic Logics (New York: Springer-Verlag, 1985), 479–506.
18. Jump up^ Peretz, Assaf, “Geometry of forking in simple theories.” J. Symbolic Logic Volume 71, Issue 1 (2006), 347–359.
19. Jump up^ Shelah, Saharon (1999). "Borel sets with large squares". Fundamenta Mathematicae 159 (1): 1–50. arXiv:math/9802134.
20. Jump up^ Makowsky J, “Compactness, embeddings and definability,” in Model-Theoretic Logics, eds Barwise and Feferman, Springer 1985 pps. 645–715.
21. Jump up^ Baldwin, John T. (July 24, 2009). Categoricity. American Mathematical Society. ISBN 978-0821848937. Retrieved February 20, 2014.
22. Jump up^ Shelah, Saharon. Introduction to classification theory for abstract elementary classes.
23. Jump up^ Hrushovski, Ehud, “Kueker's conjecture for stable theories.” Journal of Symbolic Logic Vol. 54, No. 1 (Mar., 1989), pp. 207–220.
24. Jump up^ Cherlin, G.; Shelah, S. (May 2007). "Universal graphs with a forbidden subtree". Journal of Combinatorial Theory, Series B 97 (3): 293–333. arXiv:math/0512218.doi:10.1016/j.jctb.2006.05.008.
25. Jump up^ Džamonja, Mirna, “Club guessing and the universal models.” On PCF, ed. M. Foreman, (Banff, Alberta, 2004).
26. Jump up^ Ribenboim, P. (2006). Die Welt der Primzahlen (in German) (2 ed.). Springer. pp. 242–243. doi:10.1007/978-3-642-18079-8. ISBN 978-3-642-18078-1.
27. Jump up^ Dobson, J. B. (June 2012) [2011], On Lerch's formula for the Fermat quotient, p. 15, arXiv:1103.3907
28. Jump up^ Barros, Manuel (1997), "General Helices and a Theorem of Lancret", American Mathematical Society 125: 1503–1509, JSTOR 2162098.
29. Jump up^ Franciscos Santos (2012). "A counterexample to the Hirsch conjecture". Annals of Mathematics (Princeton University and Institute for Advanced Study) 176 (1): 383–412.doi:10.4007/annals.2012.176.1.7.
30. Jump up^ Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre’s modularity conjecture (I)", Inventiones Mathematicae 178 (3): 485–504, doi:10.1007/s00222-009-0205-7 andKhare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre’s modularity conjecture (II)", Inventiones Mathematicae 178 (3): 505–586, doi:10.1007/s00222-009-0206-6.
31. Jump up^ Green, Ben (2004), "The Cameron–Erdős conjecture", The Bulletin of the London Mathematical Society 36 (6): 769–778, arXiv:math.NT/0304058,doi:10.1112/S0024609304003650, MR 2083752.

• Weisstein, Eric W., "Unsolved problems", MathWorld.
• Waldschmidt, Michel (2004). "Open Diophantine Problems". Moscow Mathematical Journal 4 (1): 245–305. ISSN 1609-3321. Zbl 1066.11030.

Books discussing unsolved problems[edit]

• 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 0-06-093558-8.
• John Derbyshire (2003). Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Joseph Henry Press. ISBN 0-309-08549-7.
• Keith Devlin (2006). The Millennium Problems – The Seven Greatest Unsolved* Mathematical Puzzles Of Our Time. Barnes & Noble. ISBN 978-0-7607-8659-8.
• Vincent D. Blondel, Alexandre Megrestski (2004). Unsolved problems in mathematical systems and control theory. Princeton University Press. ISBN 0-691-11748-9.

Books discussing recently solved problems[edit]

• Simon Singh (2002). Fermat's Last Theorem. Fourth Estate. ISBN 1-84115-791-0.
• Donal O'Shea (2007). The Poincaré Conjecture. Penguin. ISBN 978-1-84614-012-9.
• 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.

External links[edit]

• Unsolved Problems in Number Theory, Logic and Cryptography
• Clay Institute Millennium Prize
• List of links to unsolved problems in mathematics, prizes and research.
• Open Problem Garden The collection of open problems in mathematics build on the principle of user editable ("wiki") site
• AIM Problem Lists
• Unsolved Problem of the Week Archive. MathPro Press.
• The Open Problems Project (TOPP), discrete and computational geometry problems

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