Problems still waiting on a proof.

Does every problem whose solution can be verified quickly also admit a quick algorithm to find that solution? A decisive answer would redraw the map of computation, cryptography, and optimization.

Do all non-trivial zeros of the Riemann zeta function lie on the critical line Re(s) = 1/2? The hypothesis governs the finest distribution of the prime numbers.

Produce a mathematically consistent framework that unifies quantum field theory with general relativity, recovering both in the appropriate limits and making sharp, testable predictions.

Do the three-dimensional incompressible Navier–Stokes equations always admit smooth, globally defined solutions, or can singularities form from smooth initial data?

Establish a rigorous quantum Yang–Mills theory on R⁴ and prove that it has a positive mass gap: the lowest excitation sits strictly above the vacuum.

Formulate and validate a general, predictive theory of how proteins fold and misfold in physiologically relevant conditions, beyond static structure prediction alone.

Are Hodge classes on a non-singular complex projective variety algebraic, that is, rational linear combinations of cohomology classes of algebraic subvarieties?

Are there infinitely many primes p such that p + 2 is also prime? Bounded-gap breakthroughs brought us close. The infinite twin-prime statement remains open.

What is the maximum number of times the unit distance can occur among n points in the plane? The long-standing Szemerédi–Trotter ceiling of O(n^{4/3}) has not been broken.

Starting from any positive integer, repeatedly apply n → n/2 if even and n → 3n + 1 if odd. Does every trajectory eventually reach 1?

Is every even integer greater than 2 the sum of two primes? Verified for enormous ranges by machine, still unproved in general.

Consider k runners on the unit circle with distinct constant speeds. Is each runner lonely at some time (at least distance 1/k from every other), no matter the speeds?

What is the chromatic number of the plane: the smallest number of colors so that no two points exactly distance 1 apart share a color?

In any finite union-closed family of at least one nonempty set, does some element appear in at least half of the member sets?

Does every simple closed curve in the plane contain four points that form a square? The smooth and piecewise-linear cases have major partial results; the fully general Jordan curve remains open.

Can the vertices of every tree on n edges be labeled with 0…n so that the induced edge differences {1,…,n} are all distinct? Equivalent forms touch Ringel's conjecture on tree decompositions.

How many unit distances can a planar n-point set be forced to realize? The best constructions still sit only slightly above linear, far from the O(n^{4/3}) upper bound.

For every integer n ≥ 2, do there exist positive integers x, y, z such that 4/n = 1/x + 1/y + 1/z? A statement about Egyptian fraction expansions of 4/n.

In every oriented graph, is there a vertex whose out-neighborhood's second out-neighborhood is at least as large as its first? A clean statement about local expansion in orientations.

In every finite partially ordered set that is not a total order, does there exist an incomparable pair (x, y) such that the fraction of linear extensions with x before y lies between 1/3 and 2/3?

Is there a uniform bound on how often a number greater than 1 can appear in Pascal's triangle? Singmaster asked whether the multiplicity of values in binomial coefficients is bounded.

What is the maximum number of points that can be placed in the n × n grid so that no three are collinear? A construction of 2n is classical; whether 2n is always achievable for every n is open in general.