This article lists some
. See individual articles for details and sources. unsolved problems in mathematics
Millennium Prize Problems [ edit ]
Of the seven
Millennium Prize Problems set by the Clay Mathematics Institute, six have yet to be solved:
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 ]
Pompeiu problem Are
(the Euler–Mascheroni constant), + π , e π − e, π e, π/ e, π , e π , √ 2 π , e π , π 2 ln π, 2 , e e , e Catalan's constant or Khinchin's constant rational, algebraic irrational, or transcendental? 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
magic squares (sequence in A006052 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 contain two elements x and y such that the probability that x appears before y in a random linear extension is between 1/3 and 2/3?
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 ?
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. [9 ] Determine the structure of Keisler's order
[10 ] [11 ] 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? [12 ] The Stable Forking Conjecture for simple theories
[13 ] 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? [14 ] Is there a logic satisfying the interpolation theorem which is compact?
[15 ] If the class of atomic models of a complete first order theory is
categorical in the , is it categorical in every cardinal? [16 ] [17 ] Is every infinite, minimal field of characteristic zero algebraically closed? (minimal = no proper elementary substructure)
[18 ] 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?
[19 ] The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?
(Proof claimed in 2012, currently under review.) abc 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= m 2 other than n=4,5,7
Distribution and upper bound of mimic numbers
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?
Problems solved recently [ edit ]
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 ) [25 ]
Serre's modularity conjecture ( Chandrashekhar Khare and Jean-Pierre Wintenberger, 2008 ) [26 ]
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) [27 ]
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 ]
References [ edit ]
^ Weisstein, Eric W., " Pi", . MathWorld
^ Weisstein, Eric W., " e", . MathWorld
^ Weisstein, Eric W., " Khinchin's Constant", . MathWorld
^ Weisstein, Eric W., " Irrational Number", . MathWorld
^ Weisstein, Eric W., " Transcendental Number", . MathWorld
^ Weisstein, Eric W., " Irrationality Measure", . MathWorld
^ An introduction to irrationality and transcendence methods
^ Some unsolved problems in number theory
^ Shelah S, Classification Theory, North-Holland, 1990
^ Keisler, HJ, “Ultraproducts which are not saturated.” J. Symb Logic 32 (1967) 23—46.
^ Malliaris M, Shelah S, "A dividing line in simple unstable theories." http://arxiv.org/abs/1208.2140
^ Gurevich, Yuri, "Monadic Second-Order Theories," in J. Barwise, S. Feferman, eds., Model-Theoretic Logics (New York: Springer-Verlag, 1985), 479-506.
^ Peretz, Assaf, “Geometry of forking in simple theories.” J. Symbolic Logic Volume 71, Issue 1 (2006), 347-359.
^ Shelah, Saharon (1999). "Borel sets with large squares". Fundamenta Mathematicae 159 (1): 1–50. arXiv: 9802134.
^ Makowsky J, “Compactness, embeddings and definability,” in Model-Theoretic Logics, eds Barwise and Feferman, Springer 1985 pps. 645-715.
^ Baldwin, John T. (July 24, 2009). . Categoricity American Mathematical Society. ISBN 978-0821848937 . Retrieved February 20, 2014.
^ Shelah, Saharon. . Introduction to classification theory for abstract elementary classes
^ Hrushovski, Ehud, “Kueker's conjecture for stable theories.” Journal of Symbolic Logic Vol. 54, No. 1 (Mar., 1989), pp. 207-220.
^ Cherlin, G.; Shelah, S. (May 2007). "Universal graphs with a forbidden subtree". Journal of Combinatorial Theory, Series B 97 (3): 293–333. arXiv: 0512218.
^ Džamonja, Mirna, “Club guessing and the universal models.” On PCF, ed. M. Foreman, (Banff, Alberta, 2004).
^ Ribenboim, P. (2006). (in German) (2 ed.). Springer. pp. 242–243. Die Welt der Primzahlen doi: 10.1007/978-3-642-18079-8. ISBN 978-3-642-18078-1.
^ Dobson, J. B. (June 2012) , On Lerch's formula for the Fermat quotient, p. 15, arXiv: 1103.3907
^ Malliaris, M.; Shelah, S. (2012), Cofinality spectrum theorems in model theory, set theory and general topology, arXiv: 1208.5424
^ Barros, Manuel (1997), "General Helices and a Theorem of Lancret", American Mathematical Society 125: 1503–1509, JSTOR 2162098 .
^ 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.
^ Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre’s modularity conjecture (I)", Inventiones Mathematicae 178 (3): 485–504, doi: 10.1007/s00222-009-0205-7 and Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre’s modularity conjecture (II)", Inventiones Mathematicae 178 (3): 505–586, doi: 10.1007/s00222-009-0206-6 .
^ 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 .
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 ]
External links [ edit ]