Click a problem.
Read the problems below like a native. A college-level guide to what the notation means and how it works: logic, sets, numbers, calculus, and the Greek alphabet.
The hidden order of the primes.
Is checking an answer as hard as finding it?
Which shapes are built from algebra?
Do fluid equations always stay smooth?
The math behind the forces of nature.
Counting rational points on curves.
Every even number = two primes?
Infinitely many primes two apart?
The 3n+1 problem: always back to 1?
18 problems for the 21st century: each with its status.
Four prime problems Landau called unattackable: all still open.
The 1900 list that shaped modern math: each with its status today.
3-manifolds, geometrization, and what Perelman & Agol settled.
Do divisor-sum chains always stop or loop: or can one run away forever?
Number theory
Is 1+2=3 the only consecutive-power-sum coincidence?
Number theory
A distinct prime tag for every integer in a run of composites.
Number theory
Does any number have no abundancy twin?
Number theory
One polynomial congruence to certify a prime: too good to be true?
Number theory / computation
How fast can you really multiply two matrices?
Theoretical CS
Does low matrix rank force low communication cost?
Theoretical CS
A closed form for the 3D magnet: almost certainly none.
Statistical mechanics
Spread N charges on a sphere so repulsion is least.
Mathematical physics
Riemann, for every L-function at once.
Number theory
How many lattice points fit inside a circle?
Number theory
The true size of the divisor-sum error.
Number theory
Do the primes' signs behave randomly?
Number theory
Möbius is blind to orderly dynamics.
Number theory / dynamics
A primitive root for infinitely many primes?
Number theory
The grand unified theory of mathematics.
Representation theory
Every number as a few k-th powers.
Number theory
Are π's digits perfectly evenly spread?
Number theory
Is 0.91597… even irrational?
Number theory
ζ(3) is irrational: is ζ(5)?
Number theory
When do several polynomials hit primes together?
Number theory
Can φ(n) divide n−1 for a composite n?
Number theory
Is every totient value taken more than once?
Number theory
Higher signatures as homotopy invariants.
Topology
Can an ideal fluid blow up in finite time?
PDE
The most efficient foam: Kelvin's was beaten.
Geometry
A needle set in every direction: must it be full-dimensional?
Harmonic analysis
Colouring loosely-glued cliques with n colours.
Graph theory
Only two numbers are repunits in two bases?
Number theory
Share a root with every derivative: forced to be a power?
Algebra
Is 78557 the smallest Sierpiński number?
Number theory
Differences of primes always start with 1?
Number theory
Is the nth root of the nth prime always falling?
Number theory
Every even gap between primes, infinitely often?
Number theory
A sum that detects primes exactly?
Number theory
How many points force a convex polygon?
Combinatorial geometry
Cover a convex body with 2^n smaller copies?
Convex geometry
Cover every edge with cycles, each twice?
Graph theory
Colour vertices and edges with Δ+2?
Graph theory
The conjecture that would fix approximation limits.
Complexity
Polytope diameters: disproved, but how big?
Optimization
Infinitely many primes of the form 2^p − 1?
Number theory
Only five Fermat primes: or more?
Number theory
When is n! + 1 a perfect square?
Number theory
Is the Mandelbrot set locally connected?
Complex dynamics
Is the Mahler measure bounded away from 1?
Number theory
Split any set into n+1 smaller pieces? Not always.
Discrete geometry
How many spheres can touch one at once?
Discrete geometry
How large before a family must hide a sunflower?
Combinatorics
How often can a number repeat in Pascal's triangle?
Combinatorics
Is 4/n always a sum of three unit fractions?
Number theory
A critical point near every root of a polynomial?
Analysis
The master conjecture of transcendence.
Number theory
Approximating two reals unusually well, at once.
Diophantine approx.
Does one polynomial produce infinitely many primes?
Number theory
±1 matrices with orthogonal rows: for every multiple of 4?
Combinatorics
Can every runner get far from all the others at once?
Number theory
Is there always a prime between consecutive squares?
Number theory
Must some element lie in half of a union-closed family?
Combinatorics
How many colours does it take to paint the plane?
Geometry
Does every closed curve hide a square?
Topology
The biggest sofa that rounds the corner.
Geometry
Is a graph fixed by its vertex-deleted pieces?
Graph theory
A group question the axioms cannot decide.
Set theory
A brick with every diagonal a whole number: real?
Number theory
Addition can't manufacture prime powers for free.
Number theory
Is there an infinity between the integers and the reals?
Set theory
When is a polynomial map invertible?
Algebra
Does every operator hide an invariant subspace?
Analysis
Does a single odd perfect number exist?
Number theory
The Four-Colour Theorem, vastly generalized.
Graph theory
The last unconquered Poincaré.
Topology
How many limit cycles can a polynomial flow have?
Dynamical systems
Order forced in any colouring: but how much?
Combinatorics
A million-dollar generalization of Fermat.
Number theory
A line just like the reals: settled, by proving it undecidable.
Set theory / logic
A maximal quantum measurement: does it fix everything? Yes.
Functional analysis
36 officers in a square: Euler was wrong.
Combinatorics
When can almost every number be well approximated?
Number theory
Split a complete graph into copies of one tree.
Graph theory
Edge weights 1, 2, 3 that colour the vertices.
Graph theory
Must tiling cubes share a full face? Not always.
Discrete geometry
Exactly 15 convex pentagons tile the plane.
Tiling
One tile that only tiles aperiodically: found.
Tiling
Four colours suffice: the first computer-proved theorem.
Graph theory
8 and 9 are the only consecutive perfect powers.
Number theory
The densest way to stack spheres: now proven.
Geometry
The key that cracked Fermat's Last Theorem.
A college-level reader's guide. Math is a language: here is its alphabet and grammar.
| ∧ | and | both must be true |
| ∨ | or | at least one is true |
| ¬ | not | negation / opposite |
| → | implies | if the left is true, the right follows |
| ↔ | if and only if | both directions hold ("iff") |
| ∴ ∵ | therefore because | conclusion reason |
| ∎ / QED | end of proof | "thus it is shown" |
| ∀ | for all | true for every object in the set |
| ∃ | there exists | at least one object works |
| ∃! | there exists exactly one | unique |
| ∄ | there does not exist | none works |
| ∈ ∉ | is / is not an element of | membership: "3 ∈ ℕ" |
| ⊆ ⊂ | subset / proper subset | contained within |
| ∪ ∩ | union / intersection | combine / overlap |
| ∅ | empty set | the set with nothing in it |
| { x : … } | set-builder | "the set of x such that …" |
| A × B | Cartesian product | all ordered pairs (a, b) |
| ℕ | naturals | 0, 1, 2, 3, … |
| ℤ | integers | …, −2, −1, 0, 1, 2, … |
| ℚ | rationals | fractions a/b |
| ℝ | reals | the whole number line, incl. π, √2 |
| ℂ | complex | a + bi, where i² = −1 |
| = ≠ | equal / not equal | |
| ≈ | approximately equal | π ≈ 3.14159 |
| ≡ | congruent / identical | in number theory: same remainder (mod n) |
| ∝ | proportional to | scales with |
| ≤ ≥ | less/greater than or equal | |
| ≪ ≫ | much less / much greater |
| Σ | summation | add a sequence of terms |
| ∏ | product | multiply a sequence of terms |
| √ n! | root factorial | n! = n·(n−1)···1 |
| ∫ ∮ | integral | area/accumulation under a curve |
| ∂ d/dx | derivative | instantaneous rate of change |
| ∇ | gradient ("del") | direction of steepest increase |
| lim | limit | value approached, never quite reached |
| ∞ | infinity | unbounded: a concept, not a number |
| π | pi | 3.14159… ratio of circle's circumference to diameter |
| Σ σ | sigma | sum (capital) / standard deviation (lower) |
| Δ δ | delta | change / a tiny amount |
| ε | epsilon | an arbitrarily small positive number |
| λ | lambda | eigenvalue / wavelength / a rate |
| θ φ | theta, phi | angles |
| ζ | zeta | the Riemann zeta function |
| ∞ ∂ ∇ | : | see Calculus above |
| f : A → B | a function | takes an input from A, returns an output in B |
| x ↦ x² | "maps to" | the rule the function applies |
| f⁻¹ | inverse | undoes f |
| g ∘ f | composition | do f, then g |
Hilbert's 8th · Millennium Prize · open since 1859
Millennium Prize · posed 1971
Millennium Prize · 1950s
Millennium Prize
Millennium Prize
Millennium Prize · 1960s
posed 1742
ancient · modern form 1849
posed 1937
36 problems, 1955 · proven 1995–2001
Offered freely. No one claims these: if a solution is ever found, it is given to the world, with no credit asked.