Reference

Open Problems

Questions nobody can answer yet, from every corner of the book, and a few famous ones that were answered.

As long as a branch of science offers an abundance of problems, so long is it alive; a lack of problems foreshadows extinction or the cessation of independent development.

— David Hilbert, Paris, 1900, translated by Mary Winston Newson

Every exercise in this book has an answer that someone already knows. These questions do not. Each sits in the chapter that gives you the ideas to understand it, usually with something small you can try. Some are centuries old, some only a few decades. A few were settled in 2026, some of them by AI systems, and the claims are still being checked; each box says how far the checking has got.

Prelude

Conjecture Open since 1742

Goldbach’s conjecture

Every even number greater than $2$ is the sum of two primes: $4 = 2 + 2,$ $6 = 3 + 3,$ $8 = 3 + 5,$ $100 = 3 + 97,$ and so on.

Posed by Christian Goldbach, in a letter to Leonhard Euler, 1742 §0.2 How to Read Mathematics

Part I · Foundations

Open problem Open since the 1970s

Graph isomorphism in polynomial time

Is there an algorithm that decides whether two networks of $n$ dots are the same network with the dots relabelled, in a number of steps bounded by a fixed power of $n ?$

Asked since the 1970s §1.2 Equivalence and Quotients
Open problem Open since 1902

The Burnside problem for exponent 5

Take two letters $a$ and $b,$ form all words in them, and impose only the rule that $w^{5} = 1$ for every word $w .$ Is the resulting group, called $B (2, 5),$ finite?

Posed by William Burnside, 1902 §1.3 Groups
Conjecture Open since 1933

The Hadamard conjecture

For every multiple of $4$ there is a square table of $+ 1$ s and $- 1$ s of that size whose rows are pairwise orthogonal: any two different rows agree in exactly half of their places.

Posed by Raymond Paley, 1933 §1.5 Linear Algebra

Part II · Shapes

Conjecture Open since 1965

The Andrews–Curtis conjecture

If a presentation with as many relations as generators describes the trivial group, it can be turned into $⟨ x_{1}, \dots, x_{n} ∣ x_{1}, \dots, x_{n} ⟩$ by three kinds of moves: multiplying one relation by another, inverting a relation, and conjugating a relation.

Posed by James Andrews and Morton Curtis, 1965 §2.3 Homotopy
Conjecture Open since the 1960s

The smooth Poincaré conjecture in dimension 4

Every 4-dimensional manifold that is homotopy equivalent to the 4-sphere $S^{4}$ is the 4-sphere not only up to homeomorphism but up to smooth deformation: $S^{4}$ has no exotic smooth structure.

Asked since the 1960s §2.4 Manifolds and Surfaces
Conjecture Open since 1989

Kalai’s 3ᵈ conjecture

A convex polytope in $d$ dimensions that is symmetric about its centre has at least $3^{d}$ faces, counting the vertices, the edges, and so on up to the polytope itself.

Posed by Gil Kalai, 1989 §2.6 The Euler Characteristic

Part III · Homology

Conjecture Open since 1985

Halperin’s toral rank conjecture

If an $r$ -dimensional torus $(S^{1})^{r}$ acts continuously on a finite cell complex $X$ so that each point is left fixed by only finitely many of its elements, then the Betti numbers of $X$ add up to at least $2^{r} :$ $\sum_{k} b_{k} (X) \geq 2^{r} .$

Posed by Stephen Halperin, 1985 §3.3 Homology Groups
Open problem Open since 1969

The exponent of matrix multiplication

Can two $n \times n$ matrices be multiplied with roughly $n^{2 + ε}$ arithmetic operations, for every $ε > 0 ?$ In other words, is the exponent $ω$ of matrix multiplication equal to $2 ?$

Asked since 1969 §3.4 Computing Homology
Open problem Open since 1911

The inscribed square problem

Does every Jordan curve in the plane pass through the four corners of some square?

Posed by Otto Toeplitz, 1911 §3.5 Maps, Invariance, and First Triumphs
Open problem Open since the 1980s

The Jones unknot problem

Is there a knot other than the unknot whose Jones polynomial is $1 ?$

Asked since the 1980s §3.7 Knots and Links
Open problem Open since 1961

Recognising the unknot quickly

Is there an algorithm that decides whether a knot diagram with $n$ crossings shows the unknot, in a number of steps bounded by a fixed power of $n ?$

Asked since 1961 §3.7 Knots and Links

Part IV · Cohomology

Problem Solution claimed in 2026

Navier–Stokes existence and smoothness

Do the Navier–Stokes equations of an incompressible fluid in three dimensions always have smooth solutions that last for all time, starting from any smooth initial velocity? Or can the velocity become infinite at some point after a finite time?

Posed by Jean Leray, 1934; settled by OpenAI, with an unreleased AI model, 2026 §4.3 Differential Forms
Conjecture Open since 1982

The 11/8 conjecture

Every closed smooth spin 4-manifold satisfies $b_{2} \geq \frac{11}{8} ∣ σ ∣,$ where $b_{2}$ is the second Betti number and $σ$ the signature of the intersection form. (For a simply connected manifold, spin means that $Q (x, x)$ is even for every $x .)$

Posed by Yukio Matsumoto, 1982 §4.6 Poincaré Duality
Problem Solution claimed in 2026

A complex structure on the 6-sphere

Can the 6-sphere be given complex coordinates: is $S^{6}$ a complex manifold of complex dimension $3 ?$

Posed by Heinz Hopf, the 1940s; settled by Levent Alpöge, working with the AI model Claude, 2026 §4.8 Curvature and Characteristic Classes
Conjecture Open since the 1930s

Hopf’s sign conjecture

If a closed manifold $M$ of dimension $2 n$ has a metric whose curvature is negative in every direction (negative sectional curvature), then $(- 1)^{n} χ (M) > 0;$ if the curvature is only nonpositive, then $(- 1)^{n} χ (M) \geq 0 .$

Posed by Heinz Hopf, the 1930s §4.8 Curvature and Characteristic Classes

Part V · The Bigger Picture

Conjecture Open since 1950

The Hodge conjecture

On a smooth complex projective variety, every rational cohomology class of type $(p, p)$ is a rational combination of the classes of algebraic subvarieties.

Posed by W. V. D. Hodge, 1950 §5.3 Horizons

Settled

Conjectures that were proved, disproved, or shown to be beyond the reach of proof altogether: how each ended is part of the story. Claims still under review stay in the parts above.

Problem Neither provable nor disprovable

The continuum hypothesis

Is there an infinite set of real numbers that is larger than the set of natural numbers $N$ but smaller than $R ?$ Cantor guessed that there is not.

Posed by Georg Cantor, 1878; settled by Kurt Gödel and Paul Cohen, 1963 §1.1 Sets and Functions
Problem Neither provable nor disprovable

Whitehead’s problem

Call an abelian group $A$ a Whitehead group if every short exact sequence $0 \to Z \to B \to A \to 0$ splits. Every free abelian group is a Whitehead group. Is every Whitehead group free?

Posed by J. H. C. Whitehead, about 1950; settled by Saharon Shelah, 1974 §1.4 Abelian Groups and Formal Sums
Conjecture Proved in 2023

The shortest paper Möbius band

A strip of paper of width $1$ and length $λ$ can be bent into a Möbius band, without stretching, tearing or passing through itself, only if $λ > 3 \approx 1.73,$ and the bound $3$ cannot be improved.

Posed by Benjamin Halpern and Charles Weaver, 1977; settled by Richard Evan Schwartz, 2023 §2.2 Gluing Spaces Together
Conjecture Proved in 2003

The Poincaré conjecture

A closed 3-dimensional manifold in which every loop can be shrunk to a point is homeomorphic to the 3-sphere $S^{3} .$

Posed by Henri Poincaré, 1904; settled by Grigori Perelman, 2003 §2.4 Manifolds and Surfaces
Conjecture Disproved in 2013

The triangulation conjecture

Every manifold can be cut into simplices: it is homeomorphic to a simplicial complex.

Asked since the 1920s; settled by Ciprian Manolescu, 2013 §2.5 Simplicial Complexes
Conjecture Proved in 2026

The cycle double cover conjecture

In every network of dots and lines with no bridge (a line whose removal would cut the network in two), one can choose a list of cycles that uses every line exactly twice.

Posed by George Szekeres and, independently, Paul Seymour, 1973–79; settled by GPT-5.6 Sol Ultra, an AI model of OpenAI, 2026 §3.1 Cycles and Boundaries
Conjecture Disproved in 2025

Is unknotting number additive?

The unknotting number $u (K)$ is the smallest number of crossings that must be switched, in some diagram, to untie $K .$ The conjecture: $u (K_{1} # K_{2}) = u (K_{1}) + u (K_{2})$ for all knots $K_{1}$ and $K_{2} .$

Asked since 1937; settled by Mark Brittenham and Susan Hermiller, 2025 §3.7 Knots and Links
Problem Solved in 2024

The Kervaire invariant one problem

In which dimensions is there a closed framed manifold, a manifold with a consistent choice of directions at every point, whose Kervaire invariant is $1,$ so that it cannot be simplified by surgery into a sphere?

Posed by Michel Kervaire and William Browder, 1960–69; settled by Michael Hill, Michael Hopkins and Douglas Ravenel (2009), and Weinan Lin, Guozhen Wang and Zhouli Xu, 2024 §5.2 Homological Algebra

Statuses as of October 2026.