Collatz conjecture
Choose a positive whole number. Halve it if it is even; otherwise triple it and add one. Repeat. Does every starting number reach one?
Mathematics
Riemann hypothesis
There are 25 primes up to 100 and 168 up to 1,000. Their uneven counts follow a broad trend; a conjecture about the zeta function predicts how far they can stray from a refined estimate.
Birch and Swinnerton-Dyer conjecture
An elliptic curve has both rational solutions and an analytic function. The conjecture predicts that these two ways of studying it encode the same rank.
Hodge conjecture
Some topological features of a complex algebraic shape look as though they should come from smaller algebraic shapes inside it. Must they?
Yang–Mills existence and mass gap
A successful framework for particle physics still needs a rigorous four-dimensional construction with a positive gap above its vacuum energy.
Goldbach conjecture
Every even whole number greater than two appears to be the sum of two primes. A proof covering all of them is still missing.
Twin prime conjecture
Prime pairs such as 11 and 13 are separated by two. Do such pairs keep appearing without end?
Odd perfect numbers
A perfect number equals the sum of its positive proper divisors. All known examples are even; the existence of an odd one remains unsettled.
Hadwiger conjecture
A graph that needs many colors is predicted to contain a correspondingly large complete graph after edges are contracted and vertices removed.
Hadwiger–Nelson problem
How many colors are needed to color every point of the plane so that points exactly one unit apart always have different colors?
Hadamard matrix conjecture
Square matrices of plus and minus ones can have mutually orthogonal rows. The conjecture predicts one for every size divisible by four.
Union-closed sets conjecture
If a finite collection of sets contains every union of two of its members, must some element occur in at least half the sets?
Lonely runner conjecture
Runners start together on a circular track at different constant speeds. Each is predicted to eventually be well separated from all the others.
Erdős–Hajnal conjecture
Forbidding one fixed induced graph is predicted to force a large group of vertices that are all connected to each other or all disconnected.
Diagonal Ramsey numbers
Color every edge of a complete graph red or blue. How many vertices force k vertices whose connecting edges all have the same color?
Graceful tree conjecture
Every tree is predicted to admit vertex numbers whose differences across edges are all distinct and fill the required range.
Square peg problem
Must every simple closed curve in the plane contain four points that are the corners of a square?
Smooth four-dimensional Poincaré conjecture
A four-dimensional space can have the topology of a sphere. Must every smooth version be smoothly equivalent to the standard sphere?
Andrews–Curtis conjecture
A set of generators and relations may describe the trivial group in a complicated way. Can a small set of permitted moves always simplify it?
Lehmer’s Mahler measure problem
A polynomial’s roots define a measure of its size. Can integer polynomials have this measure arbitrarily close to one from above?
Littlewood conjecture
Two real numbers can each be approximated by fractions. The conjecture predicts especially good simultaneous approximations using a common denominator.
Schanuel’s conjecture
The conjecture predicts how many algebraically independent numbers must appear when independent complex numbers are paired with their exponentials.
Bombieri–Lang conjecture
Algebraic spaces of general type are predicted to have rational points confined to a smaller algebraic subset, rather than spread densely everywhere.
Hilbert’s tenth problem over the rationals
Can an algorithm always decide whether an equation with integer coefficients has a solution made of rational numbers?