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Since the Renaissance, every century has seen the solution of more mathematical problems than the century before, and yet many mathematical problems, both major and minor, still elude solution. Most graduate students, in order to earn a Ph.D. in mathematics, are expected to produce new, original mathematics. That is, they are expected to solve problems that are not routine, and which cannot be solved by standard methods.
An unsolved problem in mathematics does not refer to the kind of problem found as an exercise in a textbook, but rather to the answer to a major question or a general method that provides a solution to an entire class of problems. Prizes are often awarded for the solution to a long-standing problem, and lists of unsolved problems receive considerable attention. This article reiterates the Millennium Prize list of unsolved problems in mathematics as of October 2014, and lists further unsolved problems in algebra, additive and algebraicnumber theories, analysis, combinatorics, algebraic, discrete, and Euclidean geometries, dynamical systems, partial differential equations, and graph, group, model, number, set and Ramsey theories, as well as miscellaneous unsolved problems. A list of problems solved since 1975 also appears, alongside some sources, general and specific, for the stated problems.
Lists of unsolved problems in mathematics
Over the course of time, several lists of unsolved mathematical problems have appeared. The following is a listing of those lists.
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 Thomson problem - what is the minimum energy configuration of N particles bound to the surface of a unit sphere that repel each other with a 1/r potential (or any potential in general)?
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
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?
Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?
If the class of atomic models of a complete first order theory is categorical in the , is it categorical in every cardinal?
Is every infinite, minimal field of characteristic zero algebraically closed? (minimal = no proper elementary substructure)
Kueker's conjecture
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?
The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?
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?
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.
For background on the numbers that are the focus of this problem, see articles by Eric W. Weisstein, on pi (), e (), Khinchin's Constant (), irrational numbers (), transcendental numbers (), and irrationality measures () at Wolfram MathWorld, all articles accessed 15 December 2014.
Michel Waldschmidt, 2008, "An introduction to irrationality and transcendence methods," at The University of Arizona The Southwest Center for Arithmetic Geometry 2008 Arizona Winter School, March 15–19, 2008 (Special Functions and Transcendence), see , accessed 15 December 2014.
John Albert, posting date unknown, "Some unsolved problems in number theory" , in University of Oklahoma Math 4513 course materials, see , accessed 15 December 2014.
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.
Dobson, J. B. (June 2012) , On Lerch's formula for the Fermat quotient, p. 15, arXiv:1103.3907
Barros, Manuel (1997), "General Helices and a Theorem of Lancret", American Mathematical Society, 125: 1503–1509, JSTOR2162098.
Casazza, Peter G.; Fickus, Matthew; Tremain, Janet C.; Weber, Eric (2006). Han, Deguang; Jorgensen, Palle E. T.; Larson, David Royal (eds.). "The Kadison-Singer problem in mathematics and engineering: A detailed account". Contemporary Mathematics. Large Deviations for Additive Functionals of Markov Chains: The 25th Great Plains Operator Theory Symposium, June 7-12, 2005, University of Central Florida, Florida. 414. American Mathematical Society.: 299–355. ISBN978-0-8218-3923-2. Retrieved 24 April 2015.
Fan Chung; Graham, Ron (1999). Erdos on Graphs: His Legacy of Unsolved Problems. AK Peters. ISBN1-56881-111-X.
Croft, Hallard T.; Falconer, Kenneth J.; Guy, Richard K. (1994). Unsolved Problems in Geometry. Springer. ISBN0-387-97506-3.
Guy, Richard K. (2004). Unsolved Problems in Number Theory. Springer. ISBN0-387-20860-7.
Klee, Victor; Wagon, Stan (1996). Old and New Unsolved Problems in Plane Geometry and Number Theory. The Mathematical Association of America. ISBN0-88385-315-9.
Du Sautoy, Marcus (2003). The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics. Harper Collins. ISBN0-06-093558-8.
Derbyshire, John (2003). Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Joseph Henry Press. ISBN0-309-08549-7.
Devlin, Keith (2006). The Millennium Problems – The Seven Greatest Unsolved* Mathematical Puzzles Of Our Time. Barnes & Noble. ISBN978-0-7607-8659-8.
Blondel, Vincent D.; Megrestski, Alexandre (2004). Unsolved problems in mathematical systems and control theory. Princeton University Press. ISBN0-691-11748-9.
Mazurov, V. D.; Khukhro, E. I. (2014). "Unsolved Problems in Group Theory. The Kourovka Notebook. No. 18 (English version)". arXiv:1401.0300. {{cite arXiv}}: Unknown parameter |version= ignored (help)