Unsolved problems in mathematics

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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 Ramsey theory 2.5 General algebra 2.6 Combinatorics 2.7 Graph theory 2.8 Analysis 2.9 Group theory 2.10 Other 3 Problems solved recently 4 See also 5 Quotes 6 References 7 Books discussing unsolved problems 8 Books discussing recently solved problems 9 Resources

[edit] Millennium Prize Problems

Of the seven Millennium Prize Problems set by the Clay Mathematics Institute, the six ones 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.

[edit] Other still-unsolved problems

[edit] 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

[edit] Number theory: prime numbers

• Catalan's Mersenne conjecture
• Twin prime conjecture
• infinitely many prime quadruplets
• 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
• infinitely many Fibonacci primes
• Is every Fermat number 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

[edit] General number theory

• abc conjecture
• existence of odd perfect numbers
• existence of quasiperfect numbers
• existence of odd weird numbers
• existence of Lychrel numbers
• Proof that 10 is a solitary number
• Solving the Happy Ending problem for arbitrary n
• existence of Taxicab(5, 2, n) for n>1.

[edit] Ramsey theory

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

[edit] General algebra

• Hilbert's sixteenth problem
• Hadamard conjecture
• existence of perfect cuboids
• Existence of quadratic number fields being Euclidean but not norm-Euclidean

[edit] 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

[edit] Graph theory

• Erdős-Gyárfás conjecture
• The Hadwiger conjecture
• The Ringel-Kotzig conjecture
• 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)

[edit] Analysis

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

[edit] Group theory

• Is every finitely presented periodic group finite?
• The inverse Galois problem

[edit] Other

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

See also: List of conjectures

[edit] Problems solved recently

• Road coloring conjecture (Avraham Trahtman, 2007)
• The Angel problem (Various independent proofs, 2006)
• Stanley-Wilf conjecture (Gabor Tardos and Adam Marcus, 2004)
• Poincaré conjecture (Solution by Grigori Perelman in 2002 now confirmed)
• 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)

[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

• 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.

[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. 
• 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. 

[edit] 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. 

[edit] Resources

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

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