Unsolved problems in mathematics

From Wikipedia, the free encyclopedia

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

Contents 1 The seven Millennium Prize Problems 2 Other still-unsolved problems 3 Problems solved recently 4 See also 5 Quotes 6 References 7 Books discussing unsolved problems

[edit] The seven Millennium Prize Problems

The seven Millennium Prize Problems set by the Clay Mathematics Institute are:

• P versus NP
• The Hodge Conjecture
• The Poincaré Conjecture (a proposed proof by Grigori Perelman is under review)
• The Riemann Hypothesis
• Yang-Mills Existence and Mass Gap
• Navier-Stokes existence and smoothness
• The Birch and Swinnerton-Dyer Conjecture

[edit] Other still-unsolved problems

• 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

• Twin prime conjecture
• infinitely many Mersenne primes (Lenstra-Pomerance-Wagstaff conjecture); equivalently, infinitely many even perfect numbers
• infinitely many regular primes, is their density
• infinitely many Cullen primes
• infinitely many palindromic primes in base 10
• Is every Fermat number composite for n > 4?
• Is 78,557 the lowest Sierpinski number?
• Is 509,203 the lowest Riesel number?

• abc conjecture
• existence of odd perfect numbers
• existence of quasiperfect numbers
• existence of odd weird numbers
• Proof that the 196-algorithm does not terminate when applied the number 196
• Proof that 10 is a solitary number
• Solving the happy end problem for arbitrary n

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

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

• 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

• Erdős-Gyárfás conjecture
• The Graph isomorphism problem
• 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)

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

• Is every finitely presented periodic group finite?

• Generalized star height problem
• Invariant subspace problem
• Modeling black hole mergers

See also the list of conjectures.

[edit] Problems solved recently

• Stanley-Wilf conjecture (Gabor Tardos and Adam Marcus, 2004)
• The Poincaré Conjecture (Grigori Perelman, 2002)
• Catalan's conjecture (Preda Mihăilescu, 2002)
• Kato's conjecture (Auscher, Hofmann, Lacey, 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) (Some do not accept the proof because it is heavily based on computer computation.)

[edit] See also

• Hilbert's 23 problems
• Timeline of mathematics
• Category:Unsolved problems in mathematics

[edit] Quotes

"Problems worthy of attack

Prove their worth by fighting back."

— Piet Hein (1905–1996)

[edit] References

• 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.

[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.

List resources
Basic topic lists • Topic lists • Reference tables • Countries • People • Timelines