List of unsolved problems in mathematics explained
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations. Some problems belong to more than one discipline and are studied using techniques from different areas. Prizes are often awarded for the solution to a long-standing problem, and some lists of unsolved problems, such as the Millennium Prize Problems, receive considerable attention.
This list is a composite of notable unsolved problems mentioned in previously published lists, including but not limited to lists considered authoritative. Although this list may never be comprehensive, the problems listed here vary widely in both difficulty and importance.
Lists of unsolved problems in mathematics
Various mathematicians and organizations have published and promoted lists of unsolved mathematical problems. In some cases, the lists have been associated with prizes for the discoverers of solutions.
List | Number of problems | Number unsolved or incompletely solved | Proposed by | Proposed in |
---|
| 23 | 15 | | 1900 |
Landau's problems[1] | 4 | 4 | | 1912 |
Taniyama's problems[2] | 36 | - | | 1955 |
Thurston's 24 questions[3] [4] | 24 | - | | 1982 |
| 18 | 14 | | 1998 |
| 7 | 6[5] | | 2000 |
| 15 | <12[6] [7] | | 2000 |
Unsolved Problems on Mathematics for the 21st Century[8] | 22 | - | Jair Minoro Abe, Shotaro Tanaka | 2001 |
DARPA's math challenges[9] [10] | 23 | - | | 2007 |
Erdős's problems[11] | >850 | 588 | | Over six decades of Erdős' career, from the 1930s to 1990s | |
Millennium Prize Problems
Of the original seven Millennium Prize Problems listed by the Clay Mathematics Institute in 2000, six remain unsolved to date:[5]
The seventh problem, the Poincaré conjecture, was solved by Grigori Perelman in 2003.[12] However, a generalization called the smooth four-dimensional Poincaré conjecture—that is, whether a four-dimensional topological sphere can have two or more inequivalent smooth structures—is unsolved.[13]
Notebooks
Unsolved problems
Algebra
See main article: Algebra.
applied to a complex matrix
is at most twice the
supremum of
over the
field of values of
.
.
, a
Hadamard matrix of order
exists.
, then it has no nil
one-sided ideal other than
.
is the maximum of a finite set of minimums of finite collections of polynomials.
with
disjoint bases
, it is possible to create an
matrix whose rows are
and whose columns are also bases.
is a
simply connected semisimple algebraic group over a perfect
field of
cohomological dimension at most
, then the
Galois cohomology set
is zero.
is a commutative
regular local ring, and
are
prime ideals of
, then
\dim(R/P)+\dim(R/Q)=\dim(R)
implies
.
over
number fields
have at most some bounded number
of
-
rational points?
- Wild problems: problems involving classification of pairs of
matrices under simultaneous conjugation.
with coordinate ring
, if the
derivations of
are a
free module over
, then
is
smooth.
is a mixed
Shimura variety or semiabelian variety defined over
, and
is a subvariety, then
contains only finitely many atypical subvarieties.
Group theory
See main article: Group theory.
form a partition of
, then the finite indices of said subgroups cannot be distinct.
Representation theory
, the number of irreducible complex characters of degree not divisible by a
prime number
is equal to the number of irreducible complex characters of the
normalizer of any
Sylow
-subgroup within
.
Analysis
See main article: Mathematical analysis.
and
are spectral if and only if they tile by
translation.
of
degree
and a complex number
, is there a
critical point
of
such that
|f(z)-f(c)|\le|f'(z)||z-c|
?
- The Pompeiu problem on the topology of domains for which some nonzero function has integrals that vanish over every congruent copy
- Sendov's conjecture: if a complex polynomial with degree at least
has all roots in the closed
unit disk, then each root is within distance
from some
critical point.
- Vitushkin's conjecture on compact subsets of
with
analytic capacity
- What is the exact value of Landau's constants, including Bloch's constant?
- Regularity of solutions of Euler equations
- Convergence of Flint Hills series
- Regularity of solutions of Vlasov–Maxwell equations
Transcendental numbers and diophantine approximation
(the
Euler–Mascheroni constant),
\pi+e,\pi-e,\pie,\pi/e,\pie,\pi\sqrt{2
}, \pi^, e^, \ln\pi, 2^e, e^e,
Catalan's constant, or
Khinchin's constant rational,
algebraic irrational, or
transcendental? What is the irrationality measure of each of these numbers?
[18] [19]
transcendental?
Combinatorics
See main article: Combinatorics.
runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be at least a distance
from each other runner) at some time?
[21]
grid so that no three of them lie on a line?
size sets required for the existence of a sunflower of
sets be bounded by an exponential function in
for every fixed
?
for
[23] - The values of the Ramsey numbers, particularly
Dynamical systems
See main article: Dynamical system.
conjecture)
action on the circle either Lebesgue or atomic?
Games and puzzles
See main article: Game theory.
Combinatorial games
See main article: Combinatorial game theory.
- Sudoku:
- How many puzzles have exactly one solution?
- How many puzzles with exactly one solution are minimal?
- What is the maximum number of givens for a minimal puzzle?[29]
- Tic-tac-toe variants:
- Given the width of a tic-tac-toe board, what is the smallest dimension such that X is guaranteed to have a winning strategy? (See also Hales-Jewett theorem and nd game)[30]
- Chess:
- Go:
- What is the perfect value of Komi?
- Are the nim-sequences of all finite octal games eventually periodic?
- Is the nim-sequence of Grundy's game eventually periodic?
Games with imperfect information
Geometry
See main article: Geometry.
Algebraic geometry
See main article: Algebraic geometry.
.
constructed from a positive
holomorphic line bundle
on a
compact complex manifold
and the
canonical line bundle
of
is a smooth
algebraic surface and
is an
ample line bundle on
of degree
, then for sufficiently large
, the
Seshadri constant satisfies
\varepsilon(p1,\ldots,pr;X,L)=d/\sqrt{r}
.
to the
Galois group of
.
Covering and packing
is a
triangular number, packing
circles in an equilateral triangle requires a triangle of the same size as packing
circles
- The kissing number problem for dimensions other than 1, 2, 3, 4, 8 and 24
- Reinhardt's conjecture: the smoothed octagon has the lowest maximum packing density of all centrally-symmetric convex plane sets
- Sphere packing problems, including the density of the densest packing in dimensions other than 1, 2, 3, 8 and 24, and its asymptotic behavior for high dimensions.
- Square packing in a square: what is the asymptotic growth rate of wasted space?
- Ulam's packing conjecture about the identity of the worst-packing convex solid
- The Tammes problem for numbers of nodes greater than 14 (except 24).[33]
Differential geometry
See main article: Differential geometry.
is
.
Discrete geometry
See main article: Discrete geometry.
points can be equidistant in
spaces
Euclidean geometry
See main article: Euclidean geometry.
-by-
matrix depending on
points in
in
dimensions containing a single lattice point in its interior as its
center of mass cannot have volume greater than
in
must have a distance set of nonzero
Lebesgue measure
-dimensional sets that contain a unit line segment in every direction necessarily have
Hausdorff dimension and
Minkowski dimension equal to
?
mutually-repelling particles on a unit sphere?
Graph theory
See main article: Graph theory.
Algebraic graph theory
Games on graphs
- Graham's pebbling conjecture on the pebbling number of Cartesian products of graphs
- Meyniel's conjecture that cop number is
Graph coloring and labeling
- The 1-factorization conjecture that if
is odd or even and
respectively, then a
-
regular graph with
vertices is 1-factorable.
is
class 2 if and only if it has an
overfull subgraph
satisfying
.
Graph drawing and embedding
embeddings with bounded distortion
- Harborth's conjecture: every planar graph can be drawn with integer edge lengths[50]
- Negami's conjecture on projective-plane embeddings of graphs with planar covers[51]
- The strong Papadimitriou–Ratajczak conjecture: every polyhedral graph has a convex greedy embedding
- Turán's brick factory problem – Is there a drawing of any complete bipartite graph with fewer crossings than the number given by Zarankiewicz?[52]
- Universal point sets of subquadratic size for planar graphs[53]
Restriction of graph parameters
, what is the largest graph of diameter
such that all vertices have degrees at most
?
- Jørgensen's conjecture that every 6-vertex-connected K6-minor-free graph is an apex graph[54]
- Does a Moore graph with girth 5 and degree 57 exist?
- Do there exist infinitely many strongly regular geodetic graphs, or any strongly regular geodetic graphs that are not Moore graphs?
Subgraphs
-dimensional
hypercube graph?
-vertex tournament contain as a subgraph every
-vertex oriented tree?
[62]
-dimensional doubly-
directed hypercube graph can be routed with edge-disjoint
paths.
, can all triangles be hit by a set of at most
edges?
[63]
Word-representation of graphs
Miscellaneous graph theory
Model theory and formal languages
See main article: Model theory and formal languages.
is a simple algebraic group over an algebraically closed field.
-saturated models of a countable theory.
[77] - Shelah's categoricity conjecture for
: If a sentence is categorical above the Hanf number then it is categorical in all cardinals above the Hanf number.
- Shelah's eventual categoricity conjecture: For every cardinal
there exists a cardinal
such that if an
AEC K with LS(K)<=
is categorical in a cardinal above
then it is categorical in all cardinals above
.
[78]
, or
.
- Assume K is the class of models of a countable first order theory omitting countably many types. If K has a model of cardinality
does it have a model of cardinality continuum?
[82] - Do the Henson graphs have the finite model property?
- Does a finitely presented homogeneous structure for a finite relational language have finitely many reducts?
- Does there exist an o-minimal first order theory with a trans-exponential (rapid growth) function?
- If the class of atomic models of a complete first order theory is categorical in the
, is it categorical in every cardinal?
[83] [84] - Is every infinite, minimal field of characteristic zero algebraically closed? (Here, "minimal" means that every definable subset of the structure is finite or co-finite.)
- Is the Borel monadic theory of the real order (BMTO) decidable? Is the monadic theory of well-ordering (MTWO) consistently decidable?[85]
- Is the theory of the field of Laurent series over
decidable? of the field of polynomials over
?
- Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?[86]
- Determine the structure of Keisler's order.[87] [88]
Probability theory
See main article: Probability theory.
Number theory
See main article: articles.
See also: Number theory.
General
other than
?
?
defined over a
field
of
characteristic
has a factor in common with its first through
-th derivative, then must
be the
-th power of a linear polynomial?
- Catalan–Dickson conjecture on aliquot sequences: no aliquot sequences are infinite but non-repeating.
- Erdős–Ulam problem: is there a dense set of points in the plane all at rational distances from one-another?
- Exponent pair conjecture: for all
, is the pair
an exponent pair?
do the nontrivial zeros of all automorphic L-functions lie on the critical line
with real
?
do the nontrivial zeros of all Dirichlet L-functions lie on the critical line
with real
?
do the nontrivial zeros of the Riemann zeta function lie on the critical line
with real
?
, there is some constant
such that either
or
.
-th order in a general
algebraic number field, where
is a power of a prime.
to any base number field.
if
divides
, must
be prime?
,
\zeta(1/2+it)=o(t\epsilon)
,
\liminfnn\Vertn\alpha\Vert\Vertn\beta\Vert=0
, where
is the distance from
to the nearest integer.
has the property that the fractional parts of
are less than
for all positive integers
.
,
is true for only finitely many positive
such that
.
, there is some constant
such that, for any elliptic curve
defined over
with minimal discriminant
and conductor
, we have
|\Delta|\leqC(\epsilon) ⋅ f6+\epsilon
.
\Deltak(x)=Dk(x)-xPk(log(x))
- Dirichlet's divisor problem: the specific case of the Piltz divisor problem for
is at most
plus the length of the shortest addition chain producing
.
a
normal number (i.e., is each digit 0–9 equally frequent)?
[90]
Additive number theory
See main article: Additive number theory.
See also: Problems involving arithmetic progressions.
is an
additive basis of order
, then the number of ways that positive integers
can be expressed as the sum of two numbers in
must tend to infinity as
tends to infinity.
is the sum of two
prime numbers.
-th powers of positive integers is equal to a different sum of
-th powers of positive integers, then
.
can be represented as the sum of an odd
prime number and an even
semiprime.
- Minimum overlap problem of estimating the minimum possible maximum number of times a number appears in the termwise difference of two equally large sets partitioning the set
Algebraic number theory
See main article: Algebraic number theory.
, to assign a sequence of
natural numbers to each
real number such that the sequence for
is eventually
periodic if and only if
is
algebraic of degree
?
do not divide the class number of the maximal real
subfield of the
-th
cyclotomic field.
is within a constant multiple of
.
Computational number theory
See main article: Computational number theory.
Diophantine equations
where
, all three numbers
must share some prime factor.
the only solution to the
Erdős–Moser equation?
, there are positive integers
such that
.
to the equation
with
being positive
coprime integers and
being positive integers satisfying
.
(xm-1)/(x-1)=(yn-1)/(y-1)
where
and
.
- The uniqueness conjecture for Markov numbers that every Markov number is the largest number in exactly one normalized solution to the Markov Diophantine equation.
- Pillai's conjecture: for any
, the equation
has finitely many solutions when
are not both
.
Prime numbers
See main article: Prime numbers.
is prime if and only if
and
, if
(X-1)n\equivXn-1\pmod{n,Xr-1}
, then either
is prime or
, then it is a
primitive root modulo infinitely many
prime numbers
prime numbers between consecutive squares of prime numbers, aside from
and
.
has a positive leading coefficient, is irreducible over the integers, and has no common factors over all
where
is a positive integer, then
is prime infinitely often.
- Catalan's Mersenne conjecture: some Catalan–Mersenne number is composite and thus all Catalan–Mersenne numbers are composite after some point.
- Dickson's conjecture: for a finite set of linear forms
with each
, there are infinitely many
for which all forms are
prime, unless there is some
congruence condition preventing it.
- Dubner's conjecture: every even number greater than
is the sum of two
primes which both have a
twin.
and
,
does not divide
are the sum of two
prime numbers.
, there is a prime between
and
.
- Twin prime conjecture: there are infinitely many twin primes.
- Are there infinitely many primes of the form
?
, if any two of the three conditions
or
,
is prime, and
is prime are true, then the third condition is also true.
, there are infinitely many
prime gaps of size
.
of nonconstant
irreducible polynomials over the integers with positive leading coefficients, either there are infinitely many positive integers
for which
are all
primes, or there is some fixed divisor
which, for all
, divides some
.
?
and
simultaneously?
[93] - Does every prime number appear in the Euclid–Mullin sequence?
- What is the smallest Skewes's number?
- For any given integer a > 0, are there infinitely many Lucas–Wieferich primes associated with the pair (a, −1)? (Specially, when a = 1, this is the Fibonacci-Wieferich primes, and when a = 2, this is the Pell-Wieferich primes)
- For any given integer a > 0, are there infinitely many primes p such that ap − 1 ≡ 1 (mod p2)?[94]
- For any given integer a which is not a square and does not equal to −1, are there infinitely many primes with a as a primitive root?
- For any given integer b which is not a perfect power and not of the form −4k4 for integer k, are there infinitely many repunit primes to base b?
- For any given integers
, with and are there infinitely many primes of the form
with integer
n ≥ 1?
composite for
?
Set theory
See main article: Set theory.
Note: These conjectures are about models of Zermelo-Frankel set theory with choice, and may not be able to be expressed in models of other set theories such as the various constructive set theories or non-wellfounded set theory.
?
- Does the generalized continuum hypothesis imply the existence of an ℵ2-Suslin tree?
- If ℵω is a strong limit cardinal, is
(see
Singular cardinals hypothesis)? The best bound, ℵ
ω4, was obtained by
Shelah using his
PCF theory.
?
Topology
See main article: Topology.
-dimensional
homogeneous absolute neighborhood retract is a
topological manifold.
, is the space
collapsible?
Problems solved since 1995
Algebra
Analysis
and
conjectures, Bourgain-Tzafriri conjecture and
-conjecture)
Combinatorics
- Erdős sumset conjecture (Joel Moreira, Florian Richter, Donald Robertson, 2018)[102]
- McMullen's g-conjecture on the possible numbers of faces of different dimensions in a simplicial sphere (also Grünbaum conjecture, several conjectures of Kühnel) (Karim Adiprasito, 2018)[103] [104]
- Hirsch conjecture (Francisco Santos Leal, 2010)[105] [106]
- Gessel's lattice path conjecture (Manuel Kauers, Christoph Koutschan, and Doron Zeilberger, 2009)[107]
- Stanley–Wilf conjecture (Gábor Tardos and Adam Marcus, 2004)[108] (and also the Alon–Friedgut conjecture)
- Kemnitz's conjecture (Christian Reiher, 2003, Carlos di Fiore, 2003)[109]
- Cameron–Erdős conjecture (Ben J. Green, 2003, Alexander Sapozhenko, 2003)[110] [111]
Dynamical systems
Game theory
Geometry
21st century
- Einstein problem (David Smith, Joseph Samuel Myers, Craig S. Kaplan, Chaim Goodman-Strauss, 2023, still in peer review)[120]
- Maximal rank conjecture (Eric Larson, 2018)[121]
- Weibel's conjecture (Moritz Kerz, Florian Strunk, and Georg Tamme, 2018)
- Yau's conjecture (Antoine Song, 2018)[122] [123]
- Pentagonal tiling (Michaël Rao, 2017)
- Willmore conjecture (Fernando Codá Marques and André Neves, 2012)[124]
- Erdős distinct distances problem (Larry Guth, Nets Hawk Katz, 2011)[125]
- Heterogeneous tiling conjecture (squaring the plane) (Frederick V. Henle and James M. Henle, 2008)[126]
- Tameness conjecture (Ian Agol, 2004)[100]
- Ending lamination theorem (Jeffrey F. Brock, Richard D. Canary, Yair N. Minsky, 2004)[127]
- Carpenter's rule problem (Robert Connelly, Erik Demaine, Günter Rote, 2003)
- Lambda g conjecture (Carel Faber and Rahul Pandharipande, 2003)
- Nagata's conjecture (Ivan Shestakov, Ualbai Umirbaev, 2003)[128]
- Double bubble conjecture (Michael Hutchings, Frank Morgan, Manuel Ritoré, Antonio Ros, 2002)[129]
20th century
Graph theory
can be decomposed into
copies of any tree with
edges (Richard Montgomery,
Benny Sudakov, Alexey Pokrovskiy, 2020)
[138] [139] - Disproof of Hedetniemi's conjecture on the chromatic number of tensor products of graphs (Yaroslav Shitov, 2019)[140]
- Kelmans–Seymour conjecture (Dawei He, Yan Wang, and Xingxing Yu, 2020)[141] [142] [143] [144]
- Goldberg–Seymour conjecture (Guantao Chen, Guangming Jing, and Wenan Zang, 2019)[145]
- Babai's problem (Alireza Abdollahi, Maysam Zallaghi, 2015)[146]
- Alspach's conjecture (Darryn Bryant, Daniel Horsley, William Pettersson, 2014)
- Alon–Saks–Seymour conjecture (Hao Huang, Benny Sudakov, 2012)
- Read–Hoggar conjecture (June Huh, 2009)[147]
- Scheinerman's conjecture (Jeremie Chalopin and Daniel Gonçalves, 2009)[148]
- Erdős–Menger conjecture (Ron Aharoni, Eli Berger 2007)[149]
- Road coloring conjecture (Avraham Trahtman, 2007)[150]
- Robertson–Seymour theorem (Neil Robertson, Paul Seymour, 2004)[151]
- Strong perfect graph conjecture (Maria Chudnovsky, Neil Robertson, Paul Seymour and Robin Thomas, 2002)[152]
- Toida's conjecture (Mikhail Muzychuk, Mikhail Klin, and Reinhard Pöschel, 2001)[153]
- Harary's conjecture on the integral sum number of complete graphs (Zhibo Chen, 1996)[154]
Group theory
Number theory
21st century
- André–Oort conjecture (Jonathan Pila, Ananth Shankar, Jacob Tsimerman, 2021)[158]
- Duffin-Schaeffer conjecture (Dimitris Koukoulopoulos, James Maynard, 2019)
- Main conjecture in Vinogradov's mean-value theorem (Jean Bourgain, Ciprian Demeter, Larry Guth, 2015)[159]
- Goldbach's weak conjecture (Harald Helfgott, 2013)[160] [161] [162]
- Existence of bounded gaps between primes (Yitang Zhang, Polymath8, James Maynard, 2013)[163] [164] [165]
- Sidon set problem (Javier Cilleruelo, Imre Z. Ruzsa, and Carlos Vinuesa, 2010)[166]
- Serre's modularity conjecture (Chandrashekhar Khare and Jean-Pierre Wintenberger, 2008)[167]
- Green–Tao theorem (Ben J. Green and Terence Tao, 2004)[168]
- Catalan's conjecture (Preda Mihăilescu, 2002)[169]
- Erdős–Graham problem (Ernest S. Croot III, 2000)[170]
20th century
Ramsey theory
Theoretical computer science
Topology
- Deciding whether the Conway knot is a slice knot (Lisa Piccirillo, 2020)[177] [178]
- Virtual Haken conjecture (Ian Agol, Daniel Groves, Jason Manning, 2012)[179] (and by work of Daniel Wise also virtually fibered conjecture)
- Hsiang–Lawson's conjecture (Simon Brendle, 2012)[180]
- Ehrenpreis conjecture (Jeremy Kahn, Vladimir Markovic, 2011)[181]
- Atiyah conjecture for groups with finite subgroups of unbounded order (Austin, 2009)[182]
- Cobordism hypothesis (Jacob Lurie, 2008)[183]
- Spherical space form conjecture (Grigori Perelman, 2006)
- Poincaré conjecture (Grigori Perelman, 2002)[184]
- Geometrization conjecture, (Grigori Perelman, series of preprints in 2002–2003)[185]
- Nikiel's conjecture (Mary Ellen Rudin, 1999)[186]
- Disproof of the Ganea conjecture (Iwase, 1997)[187]
Uncategorised
2010s
- Erdős discrepancy problem (Terence Tao, 2015)[188]
- Umbral moonshine conjecture (John F. R. Duncan, Michael J. Griffin, Ken Ono, 2015)[189]
- Anderson conjecture on the finite number of diffeomorphism classes of the collection of 4-manifolds satisfying certain properties (Jeff Cheeger, Aaron Naber, 2014)[190]
- Gaussian correlation inequality (Thomas Royen, 2014)[191]
- Beck's conjecture on discrepancies of set systems constructed from three permutations (Alantha Newman, Aleksandar Nikolov, 2011)[192]
- Bloch–Kato conjecture (Vladimir Voevodsky, 2011)[193] (and Quillen–Lichtenbaum conjecture and by work of Thomas Geisser and Marc Levine (2001) also Beilinson–Lichtenbaum conjecture[194] [195] [196])
2000s
- Kauffman–Harary conjecture (Thomas Mattman, Pablo Solis, 2009)[197]
- Surface subgroup conjecture (Jeremy Kahn, Vladimir Markovic, 2009)[198]
- Normal scalar curvature conjecture and the Böttcher–Wenzel conjecture (Zhiqin Lu, 2007)[199]
- Nirenberg–Treves conjecture (Nils Dencker, 2005)[200]
- Lax conjecture (Adrian Lewis, Pablo Parrilo, Motakuri Ramana, 2005)[201]
- The Langlands–Shelstad fundamental lemma (Ngô Bảo Châu and Gérard Laumon, 2004)[202]
- Milnor conjecture (Vladimir Voevodsky, 2003)[203]
- Kirillov's conjecture (Ehud Baruch, 2003)[204]
- Kouchnirenko's conjecture (Bertrand Haas, 2002)[205]
- n! conjecture (Mark Haiman, 2001)[206] (and also Macdonald positivity conjecture)
- Kato's conjecture (Pascal Auscher, Steve Hofmann, Michael Lacey, Alan McIntosh, and Philipp Tchamitchian, 2001)[207]
- Deligne's conjecture on 1-motives (Luca Barbieri-Viale, Andreas Rosenschon, Morihiko Saito, 2001)[208]
- Modularity theorem (Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, 2001)
- Erdős–Stewart conjecture (Florian Luca, 2001)[209]
- Berry–Robbins problem (Michael Atiyah, 2000)[210]
See also
Further reading
Books discussing problems solved since 1995
- Book: Singh, Simon . Simon Singh . 2002 . Fermat's Last Theorem . Fourth Estate . 978-1-84115-791-7. Fermat's Last Theorem (book) .
- Book: O'Shea, Donal . Donal O'Shea. 2007 . The Poincaré Conjecture . Penguin . 978-1-84614-012-9.
- Book: Szpiro, George G. . George Szpiro. 2003 . Kepler's Conjecture . Wiley . 978-0-471-08601-7.
- Book: Ronan, Mark . Mark Ronan. 2006 . Symmetry and the Monster . Oxford . 978-0-19-280722-9.
Books discussing unsolved problems
- Book: Fan. Chung. Fan Chung . Graham . Ron . Ronald Graham. Erdös on Graphs: His Legacy of Unsolved Problems. Erdős on Graphs . AK Peters . 1999 . 978-1-56881-111-6.
- Book: Croft . Hallard T. . Falconer . Kenneth J. . Guy . Richard K. . Kenneth Falconer (mathematician) . Richard K. Guy . 1994 . Unsolved Problems in Geometry . Springer . 978-0-387-97506-1 . registration .
- Book: Guy, Richard K. . Richard K. Guy . 2004 . Unsolved Problems in Number Theory . Springer . 978-0-387-20860-2.
- Book: Klee . Victor . Victor Klee . Wagon . Stan . Stan Wagon . 1996 . Old and New Unsolved Problems in Plane Geometry and Number Theory . registration . The Mathematical Association of America . 978-0-88385-315-3.
- Book: du Sautoy, Marcus . Marcus du Sautoy . 2003 . The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics . Harper Collins . 978-0-06-093558-0 . registration .
- Book: Derbyshire, John . John Derbyshire . 2003 . Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics . Joseph Henry Press . 978-0-309-08549-6 . registration .
- Book: Devlin, Keith . Keith Devlin . 2006 . The Millennium Problems – The Seven Greatest Unsolved* Mathematical Puzzles Of Our Time . Barnes & Noble . 978-0-7607-8659-8.
- Book: Blondel . Vincent D. . Megrestski . Alexandre . Vincent Blondel . 2004 . Unsolved problems in mathematical systems and control theory . Princeton University Press . 978-0-691-11748-5.
- Book: Lizhen. Ji. Lizhen Ji . Yat-Sun. Poon . Shing-Tung. Yau. Shing-Tung Yau . 2013 . Open Problems and Surveys of Contemporary Mathematics (volume 6 in the Surveys in Modern Mathematics series) (Surveys of Modern Mathematics) . International Press of Boston . 978-1-57146-278-7.
- Waldschmidt . Michel . Michel Waldschmidt . 2004 . Open Diophantine Problems . Moscow Mathematical Journal . 1609-3321 . 1066.11030 . 4 . 1 . 245–305 . 10.17323/1609-4514-2004-4-1-245-305 . math/0312440 . 11845578 .
- Mazurov . V. D. . Victor Mazurov . Khukhro . E. I. . 1401.0300v6 . Unsolved Problems in Group Theory. The Kourovka Notebook. No. 18 (English version) . 1 Jun 2015. math.GR .
External links
Notes and References
- .
- Shimura . G. . Goro Shimura . Yutaka Taniyama and his time . Bulletin of the London Mathematical Society . 21 . 2 . 186–196 . 1989 . 10.1112/blms/21.2.186 .
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