The Worlds Hardest Test
Problems 1-6 defined as the Millennium Prize Problems
Problems 7-9 defined as Hilbert’s Problems
Problems 10-13 defined as Landau’s Problems
Problems 14-18 defined as Smale’s Problems
1.) P vs Np: for all problems for which an algorithm can verify a given solution quickly (that is, in
polynomial time), an algorithm can also find that solution quickly.
2.) Hodge Conjecture: for projective algebraic varieties, Hodge cycles are rational linear
combinations of algebraic cycles.
3.) Riemann Hypothesis: all nontrivial zeros of the analytical continuation of the Riemann zeta
function have a real part of 1/2.
4.) Yang-Mills Existence and Mass Gap: In physics, classical Yang–Mills theory is a generalization of
the Maxwell theory of electromagnetism where the chromo-electromagnetic field itself carries
charges. As a classical field theory, it has solutions which travel at the speed of light so that its
quantum version should describe massless particles (gluons). However, the postulated
phenomenon of color confinement permits only bound states of gluons, forming massive
particles
5.) Navier Stokes Existence and Smoothness: The Navier–Stokes equations describe the motion of
fluids and are one of the pillars of fluid mechanics. However, theoretical understanding of their
solutions is incomplete. In particular, solutions of the Navier–Stokes equations often include
turbulence, for which general solutions remains one of the greatest unsolved problems in
physics, despite its immense importance in science and engineering.
Even basic properties of the solutions to Navier–Stokes have never been proven. For the threedimensional system of equations, and given some initial conditions, mathematicians have not
yet proved that smooth solutions always exist, or that if they do exist, they have bounded
energy per unit mass. [citation needed] This is called the Navier–Stokes existence and
smoothness problem.
The problem is to make progress towards a mathematical theory that will give insight into these
equations, by proving either that smooth, globally defined solutions exist that meet certain
conditions, or that they do not always exist, and the equations break down.
6.) Birch and Swinnerton-Dyer Conjecture: The Birch and Swinnerton-Dyer conjecture deals with
certain types of equations: those defining elliptic curves over the rational numbers. The
conjecture is that there is a simple way to tell whether such equations have a finite or infinite
number of rational solutions.
7.) Extend the Kronecker–Weber theorem on abelian extensions of the rational numbers to any
base number field
8.) Describe relative positions of ovals originating from a real algebraic curve and as limit cycles of a
polynomial vector field on the plane.
9.) Further development of the calculus of variations
10.) Can every even integer greater than 2 be written as the sum of two primes?
11.) Are there infinitely many primes p such that p + 2 is prime?
12.) Does there always exist at least one prime between consecutive perfect squares?
13.) Are there infinitely many primes p such that p − 1 is a perfect square? In other words: Are there
infinitely many primes of the form n2 + 1?
14.) Extend the mathematical model of general equilibrium theory to include price adjustments
15.) find a strongly-polynomial time algorithm which for given matrix A ∈ Rm×n and b ∈ Rm decides
whether there exists x ∈ Rn with Ax ≥ b.
16.) Is one-dimensional dynamics generally hyperbolic?
17.) What are the limits of intelligence? Storage, as long as intelligence is defined as knowledge
about a specific subject.
18.) Is an Anosov diffeomorphism of a compact manifold topologically the same as the Lie group
model of John Franks?
Telling whether The questions asked are solvable or not
What makes a problem unsolvable? An unsolvable problem will be a problem whose solution runs into a
loop, the answer is impossible to prove, or the question is based upon flawed logic. Defined, in order, as
Lp, Np, and Wq