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
• infinitely many Mersenne primes (Lenstra-Pomerance-Wagstaff conjecture)
• infinitely many regular primes
• infinitely many Cullen primes
• infinitely many palindromic primes in base 10
• about e^−½ of all prime numbers are regular
• infinitely many even perfect numbers
• existence of odd perfect numbers
• existence of quasiperfect numbers
• modeling black hole mergers
• What is the lowest Riesel number?

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

Problems solved recently:

• Kepler's conjecture 1998
• Fermat's last theorem 1994
• 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". Feb 3, 2004.
• 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.
• Hallard T. Croft; Kenneth J. Falconer; Richard K. Guy (1994). Unsolved Problems in Geometry, Springer. ISBN 0387975063.
• 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.
• Florentin Smarandache (2000). Definitions, Solved and Unsolved Problems, Conjectures, and Theorems in Number Theory and Geometry, Amer Research. ISBN 187958574X.