What is the worlds hardest math problem?

Answer 1

There is probably no such thing as "The World's Hardest Math Problem", however there are very hard math problems that can be found online.

The hardest interesting math problems in the world will be one of the seven (now six) famous unsolved "Millenium" problems because many talented mathematicians have tried and failed to do them.

The Millennium problems will win you a million dollar prize if you can prove your answer. They only remain unsolved because they are really hard. Only one of these problems has been proven, the others are still open. Terence Tao, in a lecture about the prime numbers mentioned that he thought "N verse NP" would be the last one to be proven. Terrence Tao is one of the most prominent living mathematicians so I trust his opinion. N verse NP is the conjecture that it is always harder to derive an answer to a question than it is to verify it. This means that a computer program would require more steps to create an answer than to check it. If you can prove or disprove it you can win a million dollars and will be certain to get a Fields Medal!

Though we cannot prove this mathematically, we believe it is harder to derive an answer than it is to verify it, because there are less possible questions after an answer has been derived than whether or not an answer exists. That question will answer itself upon completion of verification should the answer verify. This brings about another problem. Is it harder to derive an answer or disprove an answer once it has been derived?

(Also the worlds hardest math problem is one you just can't ever figure out.)

Answer 2

ρ(∂v/∂t + v∙∆v) = -∆p + ∆∙T + f

Where v is the flow velocity, ρ (rho) is the fluid density, p is the pressure, T is the deviatoric component (Cauchy stress tensor), and f represents body force (throughout the volume of the body) and ∆ (which should be upside down) is the del or nabla operator, which denotes the gradient of a vector field.

Mathematicians have been unable to prove that solutions in three dimensions always exist or that those that do exist contain no singularity. If you can find a counter example the Clay Mathematics Institute will shell out $1 million USD for it--OR for a solution to the Navier-Stokes problem. Others are also willing to pay for a solution--making this one of the more valuable problems in mathematics and physics.

While there may be more difficult problems in mathematics, there are few more interesting or with more profound implication for the technological progress of our species.