The Unsolved Math Problems Worth $1 Million Each

By Alex Brannan on Saturday, March 15, 2014
“Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost.” —W.S. Anglin, Mathematics and History

In A Nutshell

In 2000, the Clay Mathematics Institute offered a prize for anyone who could solve one of seven of math’s biggest unsolved problems. The prize was set (and remains) at $1 million per problem. To date, only one of the seven problems has been solved.

The Whole Bushel

Founded in 1998 by a Boston businessman and his wife, the Clay Mathematics Institute (CMI) is dedicated to increasing and disseminating mathematical knowledge. The CMI is a tax-exempt charitable organization that supports the work of leading math researchers in the field. In 2000, the institute presented what they called Millennium Prize Problems, which consisted of seven of math’s most difficult problems that were unsolved at the turn of the millennium. The goal of the prize problems was to show the general public that the field of mathematics is still an open one with many unsolved problems and to recognize math achievements that hold historic magnitude.

To date, only one of the seven problems has been solved. Russian mathematician Grigori Perelman published a series of papers in 2003 claiming to have solved the Poincaré conjecture. The Poincaré conjecture hypothesizes that a three-dimensional object that is connected without any holes is a sphere. In his articles, Perelman proved Thurston’s geometrization conjecture, an extension of which was Poincaré’s conjecture. After careful scrutiny of his proof, Perelman was awarded the $1 million prize, which he refused. He also did not show up to accept the Field’s Medal, the highest honor in the mathematics field, for his work in solving the problem.

The remaining six problems run the gamut of subfields in the mathematical world. The Riemann hypothesis involves a question about prime numbers raised by German mathematician Bernhard Riemann. The distribution of prime numbers does not appear to follow any logical pattern, but Riemann proposed a function that is closely related to the frequency of prime numbers. The hypothesis states that the “interesting” solutions to the function when the function equals zero lie on a specific vertical line. 10,000,000,000 solutions have been found to fit these parameters, but it is a proof that every interesting solution fits that will solve the problem. Conversely, a proof that finds a solution that does not lie on the line, and thereby disproves the Riemann hypothesis, also earns the solver $1 million.

The P vs. NP problem is largely concerned with the field of computer science. An NP problem is one whose answer is easy to check, and a P problem is one whose answer is easy to find. The question is whether or not there exists a problem that is easy for a computer to check but impossibly hard for a computer to solve.

The Yang-Mills theory is used to describe elementary particles and is an important factor in elementary particle theory. The Yang-Mills theory, which has been laboratory tested and largely confirmed to be true, depends on what is called the mass gap. The mass gap is the idea that the mass of the lightest quantum particle must be positive. Solving this problem would mean providing a theoretical proof to the Yang-Mills theory, where the lightest particle is positive.

Mathematicians in the 20th century proposed a novel way of observing complex objects. They approximated the shapes of complicated objects by combining simple geometric building blocks. This process was incredibly helpful in the field of mathematics, but it led to a generalized technique. To fulfill the task in some cases, building blocks with no geometric interpretation had to be added. These pieces, called Hodge cycles, were explained in the Hodge conjecture as being combinations of geometric pieces. Proving or disproving the Hodge conjecture would lead to the prize.

In the 19th century, the Navier-Stokes equation was recorded. Modern mathematicians believe that this equation explains and can predict the motion of water and air. The problem is that our understanding of the equation is very small. Answering this Millennium Prize Problem simply means finding the solutions to this equation through a proof.

The Birch and Swinnerton-Dyer conjecture asserts the relation between a group of rational points and the number of points on an elliptic curve. The elliptic curve is a keystone of mathematics that shows up in many areas of the field. It was used in Andrew Wiles’ proof of Fermat’s last theorem, which, before Wiles’ resolution, was considered mathematics’ biggest unsolved problem. A proof of the Birch and Sinnerton-Dyer conjecture could thus have huge implications on the math world. The same could be said for all of these problems, where proving or disproving one could change our entire perception of mathematics, at least to some degree.

Show Me The Proof

Clay Mathematics Institute: Millennium Problems
NPR: 1 Solved. 6 Millennium Prize Math Problems Remain
Poincaré Conjecture Proved—This Time for Real