This article lists some unsolved problems in mathematics. See individual articles for details and sources. This is a list of lists of unsolved problems in various subjects: Unsolved problems in biology Unsolved problems in chemistry Unsolved problems in cognitive science Unsolved problems in computer science Unsolved problems in economics Unsolved problems in Egyptology Unsolved problems in governance Unsolved problems in mathematics Unsolved problems in medicine... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...

Millennium Prize Problems

Of the seven Millennium Prize Problems set by the Clay Mathematics Institute, the six ones yet to be solved are: The Clay Mathematics Institute (CMI) is a private, non-profit foundation, based in Cambridge, Massachusetts, and dedicated to increasing and disseminating mathematical knowledge. ... The Clay Mathematics Institute (CMI) is a private, non-profit foundation, based in Cambridge, Massachusetts. ...

• 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

Diagram of complexity classes provided that P ≠ NP. The existence of problems outside both P and NP-complete in this case was established by Ladner. ... The Hodge conjecture is a major unsolved problem of algebraic geometry. ... Unsolved problems in mathematics: Is the real part of a non-trivial zero of the Riemann zeta function always ½? In mathematics, the Riemann hypothesis (also called the Riemann zeta-hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous unsolved problems. ... It has been suggested that this article or section be merged with Yang-Mills existence and mass Gap. ... This article or section is in need of attention from an expert on the subject. ... In mathematics, the Birch and Swinnerton-Dyer conjecture relates the rank of the abelian group of points over a number field of an elliptic curve E to the order of zero of the associated L-function L(E, s) at s = 1. ...

Other still-unsolved problems

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

Goldbachs conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. ... In number theory, Goldbachs weak conjecture, also known as the odd Goldbach conjecture or the 3-primes problem, states that: Every odd number greater than 7 can be expressed as the sum of three odd primes. ... In number theory, Warings problem, proposed in 1770 by Edward Waring, asks whether for every natural number k there exists an associated positive integer s such that every natural number is the sum of at most s kth powers of natural numbers. ... The Collatz conjecture is an unsolved conjecture in mathematics. ... Ben Gilbreaths conjecture is a number theory problem about prime numbers. ...

Number theory: prime numbers

• 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?

The twin prime conjecture is a famous problem in number theory that involves prime numbers. ... A prime quadruplet is a group of four primes, consisting of two pairs of twin primes separated only by three non-primes, specifically, a multiple of 2, a multiple of 15 and another multiple of 2. ... In mathematics, a Mersenne number is a number that is one less than a power of two. ... In number theory, Lenstra, Pomerance, and Wagstaff have conjectured that not only are there an infinite number of Mersenne primes, meaning prime numbers of the form 2p − 1, but that the number of Mersenne primes with exponent p less than x is asymptotically approximated by , where γ is the... In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number. ... In mathematics, regular primes are a certain kind of prime numbers. ... In mathematics, a Cullen number is a natural number of the form n · 2n + 1 (written Cn). ... A palindromic prime is a prime number that is also a palindromic number. ... A Fibonacci prime is a Fibonacci number that is prime. ... In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form where n is a nonnegative integer. ... In number theory, a Sierpinski number is an odd natural number k such that integers of the form k2n + 1 are composite (i. ... In mathematics, a Riesel number is an odd natural number k for which the integers of the form k*2n-1 are all composite. ...

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

Unsolved problems in mathematics: For every ε > 0, does there exist a K>0 such that for every triple of coprime positive integers a+b=c, with product d of their distinct prime factors, |a|+|b|+|c| < Kd1+ε? The abc conjecture in number theory was first proposed by Joseph Oesterlé and... In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number. ... In mathematics, a quasiperfect number is a natural number n for which the sum of all its divisors (the divisor function &#963;(n)) is equal to 2n + 1. ... The term weird number also refers to a phenomenon in twos complement arithmetic. ... A Lychrel number is a natural number which cannot form a palindrome through the iterative process of repeatedly reversing its digits and adding the resulting numbers. ... In mathematics a solitary number is number which does not have any friends. Two numbers m and n are friends if and only if σ(m)/m = σ(n)/n. ... The Happy Ending problem (so named by Paul ErdÅ‘s since it led to the marriage of George Szekeres and Esther Klein) is the following statement: Theorem. ...

Ramsey theory

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

In combinatorics, Ramseys theorem states that in colouring a large complete graph (that is a simple graph, where an edge connects every pair of vertices), one will find complete subgraphs all of the same colour. ... Van der Waerdens theorem states that for any positive integers r and k there exists a positive integer N such that if the integers {1, 2, ..., N} are colored, each with one of r different colors, then there are at least k integers in arithmetic progression all of the...

General algebra

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

Hilberts sixteenth problem was posed by David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900, together with the other 22 problems. ... A Hadamard matrix is a square matrix with entries +1, -1 whose rows are mutually orthogonal. ... In mathematics, an Euler Brick, named after the famous mathematician Leonhard Euler, is a cuboid with integer edges and also integer face diagonals. ...

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

It has been suggested that Date magic square be merged into this article or section. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ... In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ...

Graph theory

• 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)

In graph theory, the unproven Erdős-Gyárfás conjecture, made by the prolific mathematician Paul Erdős and a collaborator, claims that any graph with minimum degree 3 contains a cycle whose length is a power of 2. ... In computational complexity theory, the graph isomorphism problem or GI problem is the graph theory problem of determining whether, given two graphs G1 and G2, it is possible to permute (or relabel) the vertices of one graph so that it is equal to the other. ... Unsolved problems in mathematics: How many colors are needed to color the plane so that no two points at unit distance are the same color? In mathematics, more specifically in geometric graph theory, the Hadwiger–Nelson problem asks for the minimum number of colors required to color the plane such...

Analysis

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

Schanuels conjecture is that given any set of n complex numbers which have linear independence over the rational numbers, the set (up to twice the size) has transcendence degree of at least n over the rationals. ... In mathematics, the Pompeiu problem is a conjecture in integral geometry, named for Dimitrie Pompeiu, who posed the problem in 1929, as follows. ... The Euler-Mascheroni constant is a mathematical constant, used mainly in number theory, and is defined as the limiting difference between the harmonic series and the natural logarithm: Its approximate value is γ ≈ 0. ...

Group theory

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

In mathematics, one method of defining a group is by a presentation. ... In group theory in mathematics, a periodic group is a group in which each element has finite order. ... In mathematics, the inverse Galois problem concerns whether or not we can find a rational field extension with a given Galois group. ...

Other

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

See also: List of conjectures

The generalized star-height problem in formal language theory is the open question whether all regular languages defined by regular expressions that include the complement operator can be expressed using regular expressions (possibly including the complement operator) with a limited nesting depth of Kleene stars. ... In the field of mathematics known as functional analysis, one of the most prominent open problems is the invariant subspace problem, sometimes optimistically known as the invariant subspace conjecture. ... Simulated view of a black hole in front of the Milky Way. ... This is a list of mathematical conjectures, by Wikipedia page. ...

Problems solved recently

• 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)

The blue dotted region shows where an angel of power 3 could reach The Angel problem is a question in game theory proposed by John Conway. ... The Stanley-Wilf conjecture is a conjecture in the combinatorics of permutations. ... In mathematics, the Poincaré conjecture (IPA: [])[1] is a conjecture about the characterization of the three-dimensional sphere amongst three-dimensional manifolds. ... Grigori Yakovlevich Perelman (Russian: ), born 13 June 1966 in Leningrad, USSR (now St. ... Mihăilescus theorem (formerly Catalans conjecture) is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proved in 2002 by Preda Mihăilescu. ... Preda Mihăilescu (born 1955) is a Romanian mathematician who received his education at the ETH Zürich and later did research at the University of Paderborn, Germany. ... Katos conjecture is a mathematical problem named after mathematician Tosio Kato, of University of California at Berkeley. ... In mathematics, the Langlands program is a web of far-reaching and influential conjectures that connect number theory and the representation theory of certain groups. ... Laurent Lafforgue (born November 6, 1966, in Antony, France) is a French mathematician. ... The Taniyama–Shimura theorem (also called the modularity theorem) establishes an important connection between elliptic curves, which are objects from algebraic geometry, and modular forms, which are certain periodic holomorphic functions investigated in number theory. ... In mathematics, the Kepler conjecture is a conjecture about sphere packing in three dimensional Euclidean space. ... Thomas Callister Hales is an American mathematician who provided computer-aided proof of the Kepler Conjecture that the most efficient way to pack spheres was in a pyramid shape. ... In mathematics, the Milnor conjecture was a proposal by John Milnor of a description of the Milnor K-theory (mod 2) of a general field F with characteristic coprime to 2, by means of the Galois (or equivalently étale) cohomology of F with coefficients in Z/2Z. It was proved... Vladimir Voevodsky (Russian: Владимир Воеводский) (born June 4, 1966) is a Russian mathematician. ... Pierre de Fermats conjecture written in the margin of his copy of Arithmetica proved to be one of the most intriguing and enigmatic mathematical problems ever devised. ... For the French mathematician with work in the area of elliptic curves, see André Weil. ... In complex analysis, de Branges theorem, named after Louis de Branges, formerly called the Bieberbach conjecture, after Ludwig Bieberbach, states a necessary condition on a holomorphic function to map the open unit disk of the complex plane injectively to the complex plane. ... Louis de Branges de Bourcia (born August 21, 1932 in Paris, France) is a French-American mathematician. ... Example of a four-colored map The four color theorem (also known as the four color map theorem) states that given any plane separated into regions, such as a political map of the counties of a state, the regions may be colored using no more than four colors in such... Kenneth Appel (born 1932) is a mathematician who, in 1976 with colleague Wolfgang Haken at the University of Illinois at Urbana-Champaign, solved one of the most famous problems in mathematics, the four-color theorem. ... Wolfgang Haken (born June 21, 1928) is a mathematician who specialized in topology, in particular 3-manifolds. ...

See also

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

Hilberts problems are a list of twenty-three problems in mathematics put forth by German mathematician David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900. ... A timeline of pure and applied mathematics // ca. ...

Quotes

"Problems worthy of attack

Prove their worth by fighting back."

— Piet Hein (1905–1996)

Piet Hein (December 16, 1905 - April 18, 1996) was a scientist, mathematician, inventor, author, and poet, often writing under the Old Norse pseudonym Kumbel meaning tombstone. His short poems, gruks (or grooks), first started to appear in the daily newspaper Politiken shortly after the Nazi Occupation in April 1940 under... 1905 (MCMV) was a common year starting on Sunday (link will display the full calendar). ... Year 1996 (MCMXCVI) was a leap year starting on Monday (link will display full 1996 Gregorian calendar). ...

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.

March 9 is the 68th day of the year in the Gregorian calendar (69th in leap years). ... For the Manfred Mann album, see 2006 (album). ...

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;. The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics. ISBN 0060935588. 

Books discussing recently solved problems

• Simon Singh (2002). Fermat's Last Theorem. Fourth Estate. ISBN 1841157910. 
• Donal O'Shea (2007). The Poincare 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. 

Resources