This article describes some currently unsolved problems in mathematics.

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
• 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?

Problems solved recently

• Catalan's conjecture (Preda Mihăilescu, 2002)
• Kato's conjecture (Auscher, Hofmann, Lacey, and Tchamitchian, 2001)
• Taniyama-Shimura conjecture (Wiles, Breuil, Conrad, Diamond, and Taylor, 1999)
• Kepler conjecture (Thomas Hales, 1998)
• Fermat's last theorem (Andrew Wiles, 1994)
• Bieberbach conjecture (Louis de Branges, 1985)
• Four color theorem (Appel and Haken, 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

Conjectures | Unsolved problems in mathematics | Lists of unsolved problems | Mathematics-related lists

Ungelöste Probleme der Mathematik | Problemas no resueltos de la matemática | 수학의 미해결 문제 | Problemi irrisolti in matematica | בעיה פתוחה במתמטיקה | 数学上の未解決問題 | Problemas em aberto da Matemática | Luettelo ratkaisemattomista matemaattisista ongelmista | Olösta matematiska problem