Unsolved problems in mathematics

In principle, any number of unsolved mathematical problems can be described, because the subject area of mathematics is unlimited. Nevertheless, in the history of mathematics, important unsolved problems have emerged several times, which have been recognized as important within science and whose solution has therefore been and is being worked on with particular zeal. It can also happen that the problem is in principle unsolvable (not decidable ) within the presupposed formal system .

The most efficient algorithms possible to solve mathematical problems are often sought (such as the question of determining the discrete logarithm for large numbers or the Traveling Salesman problem ), for which there is a classification according to difficulty classes in computer science ( complexity theory ). See list of unsolved problems in computer science .

Millennium Problems

→ Main article : Millennium problems

Most recently, in 2000, the Clay Institute in Cambridge, Massachusetts , presented the seven (from its point of view) most important unsolved problems in mathematics and offered prize money of one million dollars each for a published solution. So far, one of the so-called Millennium Problems has been solved when Grigori Perelman was able to verify the Poincaré conjecture through his proof of the more general geometrization of 3-manifolds in 2002 .

Hilbert problems

→ Main article : Hilbert problems

David Hilbert apparently served as a model for the Clay Institute , who on August 8, 1900, at the International Congress of Mathematicians in Paris, formulated 23 previously unsolved problems in mathematics. 13 of these problems have so far been comprehensively "solved", whereby the solution in some cases consists in the proof that a solution is impossible or the underlying question cannot be determined (see e.g. Hilbert's first problem ). There are still no satisfactory results for three of them. With some problems it turned out in the course of the further development of mathematics that the question was too narrowly defined and had to be reinterpreted, others were deliberately formulated very vaguely by Hilbert (such as the axiomatization of physics), so that they tend to point to Hilbert were considered important research fields at the time. The most prominent unsolved problem is still the Riemann Hypothesis , which is also included in the Clay List. Another known problem with the list is the Goldbach conjecture .

Smale problems

→ Main article : Smale problems

In 1998 Stephen Smale drew up a list of 18 math problems, inspired by a request from Vladimir Arnold to find a replacement for the Hilbert list for the new century. Vladimir Arnold himself is known for his mathematical problems, which were also published in a book.

Other known unsolved issues and questions

Number theory

• Collatz problem (also known as 3n + 1 problem, Hasse algorithm, Ulams problem)
• abc conjecture , one of the most important open questions of the theory of Diophantine equations in number theory, which would result in many other important theorems. One example is the Erdős-Woods conjecture , which is also open .
• Are there an infinite number of prime twins ? Or even prime quadruplets or prime number sixteen ?
• Is there always at least one prime number between and ( Legendre's conjecture )? In addition to Goldbach's conjecture, the problem of twin prime numbers and the question of whether there are an infinite number of prime numbers of the form , it is one of the Landau problems (after Edmund Landau ). ${\displaystyle n^{2}}$ ${\displaystyle (n+1)^{2}}$ ${\displaystyle p}$ ${\displaystyle p=n^{2}+1}$
• Are there odd perfect numbers ? Are there infinitely many perfect numbers?
• Are there quasi-perfect numbers , that is, natural numbers that are equal to the sum of their real divisors (divisors other than 1 and )? ${\displaystyle n}$ ${\displaystyle n}$
• Beal Conjecture by Andrew Beal (a generalization of the Fermat Conjecture , Beal offered $ 1 million for the solution)
• Can every integer be represented as the sum of three integer cubes? ${\displaystyle n\not \equiv \pm 4{\pmod {9}}}$
• Are there other values ​​for the taxicab number than for and ? ${\displaystyle \operatorname {Taxicab} (k,j,n)}$ ${\displaystyle k=3}$ ${\displaystyle j=2}$
• Gilbreath's guess
• Giuga conjecture
• Singmaster guess
• Bunjakowski conjecture
• Guess by Pillai
• Are there infinitely many Sophie Germain primes ?
• Are there Fermat numbers that are prime except for the five known cases (the first five Fermat numbers)?
• Conjecture by Polignac, see Alphonse de Polignac
• Artin conjectures (after Emil Artin ), of which there are two. One conjecture concerns the L-series connected with complex finite-dimensional representations of the absolute Galois group of a number field , which Artin had introduced. According to Richard Brauer, they are meromorphic functions of the complex variables . Artin suspected that they are holomorphic, with the possible exception of a single pole at . Artin's second conjecture concerns primitive roots of unity mod . It says that any non-square integer is a primitive root of unity for an infinite number of prime numbers and that the relative density of these prime numbers in the set of prime numbers is a rational multiple of Artin's constant. The latter conjecture was proven by Christopher Hooley , assuming the generalized Riemann conjecture. ${\displaystyle r}$ ${\displaystyle L(s,r)}$ ${\displaystyle s}$ ${\displaystyle s=1}$ ${\displaystyle p}$ ${\displaystyle a\neq -1}$ ${\displaystyle p}$ ${\displaystyle p}$
• Dirichlet's divisor problem (see also number function )
• Chowla conjecture (and related Sarnak conjecture )
• Lindelöf's conjecture
• Are there infinitely many regular prime numbers ?
• Conjecture from Andrica: (Dorin Andrica 1985, is the nth prime number). Until confirmed. ${\displaystyle {\sqrt {p}}_{n+1}-{\sqrt {p}}_{n}<1}$ ${\displaystyle p_{n}}$ ${\displaystyle n=1,3002*10^{16}}$
• Lehmer's totient problem in number theory (see the article Derrick Henry Lehmer ).
• Schanuel's conjecture , a central conjecture in the theory of transcendent numbers.
• Fermat-Catalan conjecture
• Is the Euler-Mascheroni constant irrational, is it transcendent?
• Kummer-Vandiver conjecture (by Harry Vandiver , see there) about the class number of solid bodies
• Are there infinitely many real quadratic number fields with a unique prime factorization (class number 1)? According to heuristic considerations by Henri Cohen , Hendrik Lenstra and numerical calculations, this applies to about three quarters, and there is no proof. In the case of imaginary square number fields, however, there are exactly 9 with a clear prime factorization (Gaussian class number problem, Kurt Heegner , Harold Stark , Alan Baker ).
• Brocard and Ramanujan's problem
• Sierpinski problem: what is the smallest Sierpinski number ?
• Riesel problem according to Hans Riesel : What is the smallest Riesel number ? A trickle number is an odd natural number , so that is compound for all natural numbers . ${\displaystyle k}$ ${\displaystyle k\cdot 2^{n}-1}$ ${\displaystyle n}$
• Erdős-Straus conjecture
• Conjecture by Erdős about arithmetic sequences (also conjecture by Erdős and Turan): If the set A fulfills the condition , it contains arithmetic sequences of any length. From it follows the theorem of Szeméredi and the theorem of Ben Green and Terence Tao (the case that A is equal to the set of prime numbers). ${\displaystyle \textstyle \sum _{n\in A}{\frac {1}{n}}=\infty }$
• Is there a perfect Euler brick ?
• Lonely Runner Conjecture from the field of Diophantine approximation, set up by Jörg Wills (see there)
• Littlewood conjecture : Let a point in the plane be given by real coordinates (x, y), consider its path of multiples (nx, ny) and the product of the distances to the nearest integer coordinate axes, which is in any case less than or equal to a Quarter is. In general it does not converge, therefore the limit of the infimum is considered and the conjecture is: ${\displaystyle \liminf _{n\to \infty }\ n\,|(nx-\lfloor nx\rfloor )\,(ny-\lfloor ny\rfloor )|=0}$
• Montgomery pair correlation conjecture (by Hugh Montgomery ). It provides a relationship between the distribution of the distances between the zeros of the Riemann zeta function on the critical straight line and the distribution of the eigenvalues ​​of a Hermitian random matrix .
• Beilinson conjectures and Bloch-Kato conjecture (it is to be distinguished from another Bloch-Kato conjecture proved by Vladimir Voevodsky ). They generalize the connection of special values ​​of L-functions (Dedekind's zeta function) to global invariants of number fields in Dirichlet's analytical class number formula.
• Heinrich-Wolfgang Leopoldt's problem on the non-disappearance of the p-adic regulator for any algebraic number fields.
• What necessary and sufficient condition must a number meet so that it can be represented as the sum of three cube numbers ? ${\displaystyle n\in \mathbb {Z} }$
• Conjecture from Elliott and Halberstam
• Gaussian circle problem about the error term in the estimation of the integer grid points within a circle for large radii r (it should be asymptotically of the order of magnitude ). ${\displaystyle \pi r^{2}}$ ${\displaystyle O(r^{{\frac {1}{2}}+\epsilon })}$

algebra

• In the vicinity of the Burnside problem (after William Burnside ) there are still unsolved conjectures, for example: for which natural numbers m, n is the free Burnside group finite? Here, m is the rank (number of generators) and n is the exponent (there is a smallest so that for all group elements) ${\displaystyle B(m,n)}$ ${\displaystyle n}$ ${\displaystyle g^{n}=1}$
• Hadamard's conjecture about the existence of Hadamard matrices.
• Inverse problem of the Galois theory

Combinatorics, graph theory

• Hadwiger-Nelson problem : What is the minimum number of colors required to color a plane if two points with a distance have to be colored differently? ${\displaystyle 1}$
• Hadwinger's conjecture in graph theory
• Determination of Ramsey numbers like ${\displaystyle R(5,5)}$
• Problem of determining the number of magic squares (only known for small side lengths).
• Unit distance problem by Paul Erdős : we are looking for an upper bound as sharp as possible for the number of points with unit distance from each other for n points of the plane (see unit distance graph ). ${\displaystyle u(n)}$
• The Erdős-Szekeres problem: Erdős and George Szekeres proved in 1935 ( Erdős and Szekeres theorem ) that for each there are a number of points in the plane in general position that form the corners of a convex n-gon. Erdős and Szekeres suspected that for everyone . ${\displaystyle n>3}$ ${\displaystyle N(n)}$ ${\displaystyle N(n)=2^{n-2}+1}$ ${\displaystyle n\geq 3}$
• Harborth conjecture: does every planar graph have a representation with integer edge lengths? (after Heiko Harborth )
• Erdős and Gyárfás conjecture: Every graph with degree three or higher contains a cycle with a length that is a power of two.
• For which natural numbers are there finite projective (or affine) levels of order ? For prime powers only? For ? ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n=12,15,18,26,30\ldots }$
• Reconstruction conjecture for graphs (graph reconstruction conjecture) by Stanislaw Ulam and Paul J. Kelly . Is a graph with three or more nodes uniquely determined by the subgraphs that are obtained when one node is removed? Kelly proved this in a positive sense for trees.
• Erdős-Faber-Lovász conjecture: take k complete graphs, each with exactly k nodes. Each pair of these graphs have at most one vertex in common. Then the union of these graphs can be colored with k colors.
• Conjecture about union-closed sets conjecture, Peter Frankl 1979: consider a finite family of finite sets with the property that the union of individual sets of the family is again in the family. Then there is an element that is in at least half of the sets.
• Graceful Tree Conjecture (Ringel Kotzig)
• Formula for or values ​​of the numbers N (r, l) (van der Waerden numbers) in van der Waerden's theorem
• Total discoloration assumption (Behzad, Vizing)

Geometry, topology

• Novikov's conjecture from SP Novikov in topology. The conjecture says the homotopy invariance of the higher signatures (generalizations of the signature ) of a manifold.
• Baum-Connes conjecture by Paul Frank Baum and Alain Connes on the topological characterization of the space of irreducible unitary representations of a group (connected to the K-theory of operator algebras in non-commutative geometry). From it follows the Novikov conjecture.
• Carathéodory conjecture (according to Constantin Carathéodory ) in differential geometry: every convex, closed, sufficiently smooth surface in three-dimensional Euclidean space has at least two umbilical points . Examples are the sphere in which all points are umbilical points and the elongated ellipsoid of revolution with exactly two umbilical points. In 1940 Hans Ludwig Hamburger gave proof of analytical surfaces.
• Weinstein conjecture (by Alan Weinstein ): every Reeb vector field in contact manifolds has closed orbits (see contact geometry ).
• Toeplitz conjecture ( Otto Toeplitz 1911): is there an inscribed square for every closed Jordan curve (that is, all corners lie on the curve)? For special cases like piecewise analytic curves (like polygons, Arnold Emch 1916) or convex curves, this is known to be the case. The general case is open.
• Closest packing of spheres in higher dimensions are mostly unknown (the three-dimensional case is the Kepler conjecture ).
• Kiss numbers in different dimensions.
• The sausage conjecture, see theory of finite packing of spheres .
• Is the unknot the only knot whose Jones polynomial is the same ? It is generally assumed that this is the case (Knot Detection Conjecture), but this is not the case for entanglements . In knot theory there are many other easy-to-pose but unsolved problems. ${\displaystyle 1}$
• There are different algorithms to determine whether a node is trivial (untangable) or not, is there also a polynomial time algorithm?
• Kakeya conjecture : has a Besikowitsch set (it contains a unit segment in every orientation) in the Hausdorff dimension ? (see Sōichi Kakeya , open to ) ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle n}$ ${\displaystyle n\geq 3}$
• In a 1982 list of 24 problems about 3-manifolds by William Thurston , all but one have been solved: Are there two hyperbolic 3-manifolds whose volumes are not rationally related to one another? In general, little is known about the volume of hyperbolic 3-manifolds.
• Hopf conjecture (there are several of them): a compact symmetric space of rank greater than 1 cannot have a Riemannian metric with a positive section curvature. This is especially true for . ${\displaystyle S^{2}\times S^{2}}$
• Hilbert-Smith conjecture (after Hilbert and Paul A. Smith ): is a locally compact topological group with a true group effect in a topological manifold a Lie group? (seen by some as the correct formulation of Hilbert's 5th problem)
• Besides the circle, the 6-sphere is the only sphere on which almost complex structures exist. It is unclear whether complex structures exist on the 6-sphere or there is no proof that this is not the case.
• Falconer's conjecture , let S be a compact set in Euclidean d-dimensional space with Hausdorff dimension greater than , then the set of distances between points in S has positive Lebesgue measure. ${\displaystyle {\frac {d}{2}}}$
• Erdős-Ulam problem , is there a dense subset of the level whose points all have rational distances from one another?

Analysis, dynamic systems

• Conjecture by Mark J. Ablowitz , A. Ramani, Harvey Segur about the applicability of the inverse scattering transformation to nonlinear partial differential equations of the evolution type, namely that these have reductions to ordinary nonlinear differential equations with Painlevé property .
• Is the Mandelbrot set locally connected everywhere? The problem is one of the main problems of complex dynamics (MLC conjecture). A positive answer would imply that the amount of Mandelbrot is hyperbolic.
• Conjecture by Alexandre Eremenko : Let be a whole transcendent complex function, then every connected component of the escape set E (escaping set, that is, the one for the iteration ) is unbounded. In a tightened version, it is assumed that there is an arc in E that connects with . ${\displaystyle f}$ ${\displaystyle z\in \mathbb {C} }$ ${\displaystyle f^{n}(z)\to \infty }$ ${\displaystyle z\in E}$ ${\displaystyle \infty }$
• Conjecture by Berry and Tabor ( Michael Berry , Michael Tabor 1977): In the generic case of quantum chaos , quantum dynamics of the geodetic flow on compact Riemann surfaces, the energy eigenvalues ​​of the associated Hamilton function behave like independent random variables, if the underlying classical system is exactly integrable .
• Lehmer problem or Mahler measure problem by Lehmer (after Derrick Henry Lehmer ) in analysis.
• Pompeiu problem of analysis, after Dimitrie Pompeiu (see there).
• One problem that Ian Stewart has added to his list of unsolved problems is whether the “Autobahn” is an attractor in a cellular automaton called Langton's Ant (given any initial conditions).
• Problem of invariant subspaces (invariant subspace problem). It is a whole complex of questions, of which a series of partial results and open questions are known, depending on the choice of the underlying room or operator type. The question is whether an operator T has a nontrivial invariant subspace W in an infinitely dimensional space H (often Hilbert or Banach spaces) ( ). Per Enflo found a counterexample for Banach spaces . For finite dimensional vector spaces, however, the existence of invariant subspaces of linear operators (matrices) is the rule (see subspace ). ${\displaystyle T(W)\subseteq W}$
• The HRT presumption (based on Christopher Heil, Jay Ramanathan, Pankaj Topiwala 1996). Let and be a square - integrable complex-valued function that does not vanish identically (i.e. not for all ). Then the conjecture claims that they are linearly independent. It has only been proven for special configurations . The assumption applies if they are collinear, if they lie on a grid (and thus for up to three arbitrary points in the plane). It is already open for the case of four points in any position in the plane. However, it was proven for special configurations of four points (so-called (2.2) configurations, two of the points each lie on two different straight lines) by Ciprian Demeter and Alexandru Zaharescu. There are also variants that consider special function classes. ${\displaystyle \Lambda ={(a_{j},b_{j})\in \mathbb {R} ^{2},j=1,\dots ,N}}$ ${\displaystyle f(x)}$ ${\displaystyle f\in L^{2}(\mathbb {R} )}$ ${\displaystyle f(x)=0}$ ${\displaystyle x}$ ${\displaystyle f(x-a_{j})e^{2\,\pi \,i\,b_{j}}}$ ${\displaystyle \Lambda }$ ${\displaystyle {a_{j},b_{j}}}$
• Sendow's hypothesis (also Ilief's hypothesis) from function theory (after the Bulgarian mathematician Blagowest Sendow , 1958). Let the roots of a polynomial of the nth degree ( ) be in the complex unit disk, then each root has a maximum distance of 1 from a critical point of the polynomial. Proven for . ${\displaystyle n\geq 2}$ ${\displaystyle n<11}$

Algebraic Geometry

• Nagata problem about the degree of a plane algebraic curve with given points and multiplicities (see Masayoshi Nagata )
• Jacobi conjecture by Ott-Heinrich Keller (see there). Let be a polynomial map whose Jacobide terminant does not vanish anywhere. Is the mapping then bijective ? She is one of the problems on Stephen Smale's list. ${\displaystyle f\colon \mathbb {C} ^{n}\to \mathbb {C} ^{n}}$
• Standard conjectures for algebraic cycles by Alexander Grothendieck about the connection between algebraic cycles and Weil cohomology theories in algebraic geometry, with which Grothendieck originally hoped in the 1960s to fully prove the Weil conjectures (including the Riemann conjecture) and to construct a theory of pure motifs which has proven to be too difficult to this day. Many of the standard conjectures would derive from the Hodge conjecture (one of the Millennium Problems) and its arithmetic analog, the Tate conjecture (from John T. Tate ).
• André-Oort conjecture about Shimura varieties in arithmetic geometry (after Yves André , Frans Oort ). Proven in special cases (including Emmanuel Ullmo , Andrei Yafaev , Bas Edixhoven , Jonathan Pila , Jacob Tsimerman ).
• Resolution of singularities in algebraic geometry. The case of the characteristic 0 of the underlying bodies was solved by Heisuke Hironaka , the case of the finite characteristic is open in most cases (four and more dimensions) (for dimensions two and three Shreeram Abhyankar ). ${\displaystyle p}$

Other areas

A series of open problems in mathematical physics was compiled by Barry Simon in 1984 ( Simon Problems , updated 2000).

Solutions to famous problems

• 1974: last of the Weil conjectures proven
• 1977: four-color problem
• 1983: Mordell's presumption
• 1985: Bieberbach's conjecture
• 1995: Fermat's conjecture
• 1998: Kepler's conjecture
• 2002: Catalan conjecture
• 2002: Poincaré conjecture

Others

There are known problem compilations for various branches of mathematics, for example by Robion Kirby for the geometry and topology of low-dimensional manifolds, Shing-Tung Yau for differential geometry (1982) or the book by Richard K. Guy on unsolved problems in elementary number theory. The Hungarian mathematician Paul Erdős is known for numerous problems (some are listed above), for the solution of which he often spent small and large sums of money himself. The Polish maths school of the interwar period is also known for its orientation towards problems, collected for example in the Scottish Book .

At the invitation of Hendrik Kloosterman at the International Congress of Mathematicians in Amsterdam in 1954, John von Neumann gave a lecture on unsolved problems in mathematics, which was supposed to give a similar overview as Hilbert in 1900 at the congress in Paris. Von Neumann dealt with problems from his own research area, in particular operator algebras, fundamentals of quantum mechanics and the associated probability theory and logic. The lecture was never published (neither in the lecture volumes on the ICM 1954 nor in von Neumann's collected works). He saw the development of a theory of unlimited operators in Hilbert spaces as a central problem with regard to the justification of quantum mechanics. He gave an overview of his classification of von Neumann algebras and explained why he saw type algebras ${\displaystyle II_{1}}$ as promising candidates for a mathematical theory of quantum mechanics (something that is not otherwise found in his publications and in the estate and in which he also found the other historical development did not follow).

Classic "unsolved" problems of geometry

For centuries there were also some famous unsolved problems ( constructions ) in geometry , a branch of mathematics . These are also called the " Classical Problems of Ancient Mathematics ". It was not until 1882 (proof of the impossibility of squaring the circle ) that the last of these “unsolved” geometric problems could also be recognized as an “impossible to solve” problem. The key to the solution was to reduce geometric to algebraic problems.

Two other classic problems that long preoccupied mathematicians were the proof of the axiom of parallels from the other axioms of Euclidean geometry, which led to the development of non-Euclidean geometries in which the axiom does not hold, and the question of the solvability of equations higher than the fourth degree by radicals , which was recognized as generally unsolvable by Galois theory and the work of Niels Henrik Abel .

See also

• List of unsolved problems in computer science
• List of unsolved problems in physics

literature

• JM Abe, S. Tanaka: Unsolved problems in mathematics for the 21st century, IOS Press 2001
• Vincent Blondel, Alexandre Megrestski: Unsolved Problems in Mathematical Systems and Control Theory, Princeton UP, 2009
• Fan Chung , Ronald Graham : Erdős on Graphs: His Legacy of Unsolved Problems, AK Peters, 1999
• Hallard T. Croft, Kenneth J. Falconer , Richard K. Guy: Unsolved Problems in Geometry, Springer 2013
• Victor Klee , Stan Wagon: Old and new unsolved problems in number theory and the geometry of the plane , Birkhäuser, Basel 1997, ISBN 3-7643-5308-2 ( Zentralblatt review )
• Pierre Basieux: The Top Seven Mathematical Guesswork . rororo, 2004, ISBN 3-499-61932-6
• Richard K. Guy : Unsolved problems in number theory (3rd edition), Springer-Verlag, New York 2004, ISBN 0-387-20860-7 (English; Zentralblatt review )
• Wilfred Hulsbergen: Conjectures in arithmetic algebraic geometry: a survey , Vieweg 1992
• Elliott Pearl: Open problems in topology , Elsevier, 2007, ISBN 0-444-52208-5 (English; Zentralblatt review )
• Daniel Shanks : Solved and unsolved problems in number theory , Chelsea 1978
• Ian Stewart : The Final Riddles of Mathematics , rororo 2015
• Heinrich Tietze : Solved and unsolved mathematics problems from old and new times. 14 lectures for laypeople and friends of mathematics , 2 volumes, DTV 1982, also 7th edition, Beck Verlag, Munich 1980
• John Forbes Nash , jr., Michael Th. Rassias (eds.): Open Problems in Mathematics , Springer 2016

Individual evidence

1. ↑ Wladimir Arnold: Arnold's problems, 2nd ed., Springer 2004
2. ^ Weisstein, Eric W .: Landau's Problems, MathWorld
3. ↑ Artin`s Constant, Mathworld
4. ^ Clifford Pickover, Math Book, Sterling Publ. 2012, p. 482
5. ^ Fermat Catalan Conjecture, Mathworld
6. ^ WWJ Hulsbergen, Beilinson Conjectures, Encyclopedia of Mathematics
7. ^ Peter Schneider, The Beilinson conjectures (pdf)
8. ↑ Guido Kings: The Bloch-Kato conjectures on special values ​​of L-functions , Journal de théorie des nombres de Bordeaux, 20, 2003, 179-198
9. ^ Lackenby, Elementary Knot Theory, 2016, Arxiv
10. ↑ Thurston, Three-dimensional manifolds, Kleinian Groups and hyperbolic geometry, Bull. AMS, Volume 6, 1982, pp. 357-379.
11. ↑ Stefan Friedl, Thurston's Vision and the Virtual Fibering Theorem for 3-Manifolds, Annual Report DMV, 2014, Issue 4, pdf
12. ↑ Ian Stewart, The Last Riddles of Mathematics, rororo 2015, chapter 17
13. ↑ salvation Ramanathan, Topiwala, linear independence of time-frequency trans lates, Proc. At the. Math. Soc., Vol. 124, 1996, p. 2787, pdf
14. ↑ Demeter, Zaharescu, Proof of the HRT conjecture for (2.2) configurations, Arxiv 2010
15. ↑ Eric Weisstein: Simon's Problems
16. ↑ Kirby, Problems in low dimensional manifold theory, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), pp. 273-312
17. ^ Yau, Problem Section, in: Yau (Ed.), Seminar on Differential Geometry, Princeton UP, 1982, pp. 669-706.
18. ↑ Miklós Rédei, "Unsolved Problems in Mathematics", from Neumann's Address International Congress of Mathematicians, September 2-9, Mathematics to the Amsterdam, 1954 , Mathematical Intelligencer 1999, No. 4. A typescript of the speech is in the von Neumann archive of the library of Congress.
19. ↑ Five Millennium problems are dealt with (P = NP, Navier-Stokes equation, Riemann hypothesis with an essay by Alain Connes, Hodge hypothesis, Birch-Swinnerton-Dyer hypothesis), Montgomery's pair correlation conjecture, generalized Fermat equations such as that of Andrew Beal, the plateau problem , the unknot problem, the question of better applicability of cooperative game theory in economics, the Nowikow conjecture and related problems (Baum-Connes), the Goldbach conjecture, Hadwiger's conjecture, the Hadwiger-Nelson conjecture Problem, the Erdős-Szekeres problem, the Erdős unit distance problem and the discrete logarithm problem .

Web links

• Unsolved Problems, Mathworld
• www.zeit.de - on the problem of a proof with a computer With further links on the subject
• Website Clay Institute of the seven Millennium problems (English)
• Alan Coley, Open Problems in Mathematical Physics, 2017
• DARPA Mathematics Challenges, 2007
• Open problem garden