Mathematics · Curated list
Riemann hypothesis, Goldbach's conjecture, Seven Bridges of Königsberg and 27 more, ranked by how much of the world has written about each one.
Most written-about first. Each description summarises the conjecture or problem's Wikipedia article, which its name links to.
1
Conjecture on zeros of the zeta function
A conjecture about where the Riemann zeta function vanishes: at the negative even integers, and otherwise only at complex numbers whose real part is ½. Many regard it as the most important open problem of pure mathematics.
Wikidata · Photo: AstroOgier, CC BY-SA 4.0
2
Even integers as sums of two primes
Among the oldest and most famous open questions of number theory, and of mathematics as a whole. It proposes that any even natural number above 2 can be written as two primes added together.
Wikidata · Photo: Adam Cunningham and John Ringland, CC BY-SA 3.0
3
Classic problem in graph theory
A historical puzzle: could someone walk through the city of Königsberg crossing each of its bridges once and only once? Leonhard Euler set the question out mathematically and proved it impossible in 1736, laying the groundwork for graph theory and anticipating the idea of topology.
Wikidata · Photo: Bogdan Giuşcă, CC BY-SA 3.0
4
Problem of constructing equal-area shapes
A geometry problem going back to Greek mathematics: using only a compass and a straightedge, in finitely many steps, build a square whose area equals that of a given circle.
Wikidata · Photo: Original PNG by Plynn9 ; SVG by Alexei Kouprianov, Public domain
Theorem in geometric topology
A theorem of geometric topology, a field of mathematics, that characterises the 3-sphere.
6
Probability puzzle
A probability puzzle and brain teaser loosely modelled on the American TV game show Let's Make a Deal, named after its first host, Monty Hall. Steve Selvin first posed it in a 1975 letter to the American Statistician.
Unsolved problem in computer science
One of the great open questions of theoretical computer science. Put simply, it asks whether every decision problem whose proposed yes-answer can be checked quickly can also be answered quickly.
8
Probability of shared birthdays
A question of probability theory: among n people chosen at random, how likely is it that two or more have the same birthday? Its surprising answer, known as the birthday paradox, is that with just 23 people the chance rises above 50%.
Wikidata · Photo: Umberto NURS, CC BY-SA 4.0
9
Open problem on 3x+1 and x/2 functions
Among the best-known open problems in mathematics. It asks whether applying two simple arithmetic steps over and over will always bring any positive integer down to 1.
Construction of an angle equal to one third a given angle
The task of constructing an angle one third the size of any given angle with just two tools, a compass and an unmarked straightedge. It is one of the classic construction problems of ancient Greek mathematics.
11
Ancient geometric construction problem
An ancient problem of geometry, also called the Delian problem: starting from the edge of one cube, construct the edge of another cube with twice the volume.
Wikidata · Photo: Petrus3743, CC BY-SA 4.0
Sum of inverse squares of natural numbers
A problem of mathematical analysis, with a bearing on number theory, about an infinite sum of inverse squares. Pietro Mengoli first raised it in 1650; Leonhard Euler solved it in 1734, and his solution was read to the Saint Petersburg Academy of Sciences on 5 December 1735.
Mathematical problem in number theory
A question of number theory: for each natural number k, is there a positive integer s such that any natural number can be written as a sum of no more than s k-th powers? For instance, 4 squares always suffice, as do 9 cubes and 19 fourth powers.
Conjecture in number theory
A number-theory conjecture that came out of a 1985 discussion between Joseph Oesterlé and David Masser. It concerns three relatively prime positive integers a, b and c for which a + b = c.
Theorem about consecutive perfect powers
Also called Mihăilescu's theorem, this number-theory result was conjectured by Eugène Charles Catalan in 1842 and proved by Preda Mihăilescu of Paderborn University in 2002. It concerns consecutive perfect powers, like 2³ and 3², whose values, 8 and 9, sit side by side.
Four basic unsolved problems about prime numbers
Four basic questions about prime numbers that Edmund Landau set out at the International Congress of Mathematicians in 1912. In his speech he described them as "unattackable at the present state of mathematics"; they now carry his name.
Geometry problem about finding touching circles
A problem of Euclidean plane geometry: construct the circles that touch three given circles in a plane. Apollonius of Perga posed it and solved it in his book Ἐπαφαί; the book is lost, but an account of his results by Pappus of Alexandria, from the 4th century AD, survives.
18
Unsolved problem in geometry
A central open problem of algebraic and complex geometry. For a complex algebraic variety that is non-singular, it ties the variety's algebraic topology to the subvarieties it contains.
Unproved conjecture in mathematics
An open problem of number theory about the rational solutions of the equations that define an elliptic curve. It is widely seen as among the hardest problems in mathematics.
Conjecture about prime numbers, proof under review
A proposition of number theory, which also goes by the 3-primes problem, the ternary Goldbach problem and the odd Goldbach conjecture: any odd number above 5 can be written as three primes added together.
Impossible task in computing
A challenge that David Hilbert and Wilhelm Ackermann posed in 1928, in mathematics and computer science: find an algorithm that, given any statement, says "yes" when the statement holds universally (in every structure) and "no" when it does not.
Question in geometric probability
A question of probability theory first raised by Georges-Louis Leclerc, Comte de Buffon, in the 18th century. A needle is dropped on a floor of parallel wooden strips, all equally wide: how likely is it to land across a line between two strips?
Disproved conjecture in number theory
A conjecture of number theory, connected to Fermat's Last Theorem, that has since been disproved. Leonhard Euler put it before the St. Petersburg Academy of Sciences in 1778.
24
Math theorem about sphere packing
A theorem about packing spheres in three-dimensional Euclidean space, named after Johannes Kepler, the 17th-century mathematician and astronomer. It holds that when equal spheres fill space, no arrangement beats the average density of cubic close packing and hexagonal close packing.
Wikidata · Photo: Original: Blotwell Derivative work: User:AzaToth, GPL
25
There is a prime between any two square numbers
A conjecture by Adrien-Marie Legendre: for every positive integer n, some prime lies between n² and (n + 1)². It is one of Landau's problems on primes (1912), and one of many open questions about how primes are spaced.
Unsolved geometry question on moving a sofa through a 90° angle
A two-dimensional, idealised version of the everyday problem of moving furniture. It asks for the largest-area rigid shape that can be carried round an L-shaped corner whose arms are of unit width; that largest area is called the sofa constant.
27
Mathematical puzzle of avoiding crossings
A mathematical puzzle, also known as water, gas and electricity: link three houses to three utility companies on a flat surface without any connection crossing another. Henry Dudeney, setting it in the early 20th century, already called it an old problem.
Mathematical problem involving optimal stopping theory
A scenario from optimal stopping theory, studied widely in applied probability, statistics and decision theory. Its other names include the googol game, the best choice problem, the fussy suitor problem, the sultan's dowry problem and the marriage problem.
Millennium Prize Problem
An open question in mathematics since the early 20th century: in three-dimensional Euclidean space, and from given starting conditions with perhaps an external force, must the Navier–Stokes equations always have solutions that are smooth? They are partial differential equations, taken together as a system, that describe how a fluid moves.
Millennium Prize Problem
An open problem of mathematical physics and mathematics. It is one of the seven Millennium Prize Problems set by the Clay Mathematics Institute, which offers US$1,000,000 for a solution.
Every Wikidata item that is a mathematical conjecture or a mathematical problem, solved or not, and that has its own English Wikipedia article, ranked by the number of Wikimedia sites with a page about it — Wikipedia's language editions, mostly, and sister projects such as Wikiquote. That counts how many communities independently thought it worth describing, and nobody can buy a place on it. The top 30 are shown.
Wikidata files these under this list's query, but they are not what the list is about:
These belong on the list but Wikidata does not file them where the query looks:
Fetched from Wikidata on 2026-09-25. Each description is this site's summary of the entry's Wikipedia article, which its name links to: the wording is ours, the facts are Wikipedia's, licensed CC BY-SA 4.0. To correct an entry, correct it there; the next refresh carries the change.
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