List of unsolved problems in mathematics

This article describes some currently unsolved problems in mathematics.

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

• P versus NP
• The Hodge Conjecture
• The Poincaré Conjecture
• The Riemann Hypothesis
• Yang-Mills Existence and Mass Gap
• Navier-Stokes Existence and Smoothness
• The Birch and Swinnerton-Dyer Conjecture

Other still-unsolved problems:

• Twin prime conjecture
• Number of Magic squares
• Gilbreath's conjecture
• Goldbach's conjecture
• Goldbach's weak conjecture
• Perfect cuboid
• Hilbert's sixteenth problem
• Generalized star height problem
• Collatz conjecture
• Schanuel's conjecture
• abc conjecture
• infinitely many Mersenne primes (Lenstra-Pomerance-Wagstaff conjecture); equivalently, infinitely many even perfect numbers
• infinitely many regular primes
• infinitely many Cullen primes
• infinitely many palindromic primes in base 10
• all Fermat number is composite for n>4.
• there are infinitely/finitely many Fermat number that are prime/composite number.
• Lehmer's conjecture
• about e^−½ of all prime numbers are regular
• existence of odd perfect numbers
• existence of quasiperfect numbers
• modeling black hole mergers
• Is γ (the Euler-Mascheroni constant) irrational?
• What is the lowest Riesel number?

See also those problems listed as conjectures (list of conjectures).

Problems solved recently:

• Catalan's conjecture 2002
• Kato's conjecture 2001
• Taniyama-Shimura conjecture 1999
• Kepler conjecture 1998
• Fermat's last theorem 1994
• Bieberbach conjecture 1985
• Four color theorem 1977

See also: Hilbert's 23 problems, Timeline of mathematics

Quotes

"Problems worthy of attack

Prove their worth by fighting back."

— Piet Hein (1905–1996)

External links

• Winkelmann, Jörg, "Some Mathematical Problems". Mar 9, 2006.
• List of links to unsolved problems in mathematics, prizes and research.

Books discussing unsolved problems

• Fan Chung; Ron Graham (1999). Erdos on Graphs : His Legacy of Unsolved Problems. AK Peters. ISBN 156881111X.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Hallard T. Croft; Kenneth J. Falconer; Richard K. Guy (1994). Unsolved Problems in Geometry. Springer. ISBN 0387975063.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Richard K. Guy (2004). Unsolved Problems in Number Theory. Springer. ISBN 0387208607.
• Victor Klee; Stan Wagon (1996). Old and New Unsolved Problems in Plane Geometry and Number Theory. The Mathematical Association of America. ISBN 0883853159.{{cite book}}: CS1 maint: multiple names: authors list (link)