If you can solve one of these 6 major math problems, you'll win a $1 million prize

Reflecting the importance of the problems, the Institute offered a $1 million prize to anyone who could provide a rigorous, peer-reviewed solution to any of the problems.

While one of the problems, the Poincare Conjecture, was famously solved in 2006 (with the mathematician who solved it, Grigori Perelman, equally famously turning down both the million dollar prize and the coveted Fields Medal), the other six problems remain unsolved.

Here are the six math problems so important that solving any one of them is worth $1 million.

P vs NP

Read the Clay Mathematics Institute's official description of P vs NP here.

The Navier-Stokes equations

Read the Clay Mathematics Institute's official description of the Navier-Stokes equations here.

Yang-Mills theory and the quantum mass gap

Math and physics have always had a mutually beneficial relationship. Developments in mathematics have often opened new approaches to physical theory, and new discoveries in physics spur deeper investigations into their underlying mathematical explanations.

Quantum mechanics has been, arguably, the most successful physical theory in history. Matter and energy behave very differently at the scale of atoms and subatomic particles, and one of the great achievements of the 20th century was developing a theoretical and experimental understanding of that behavior.

One of the major underpinnings of modern quantum mechanics is Yang-Mills theory, which describes the quantum behavior of electromagnetism and the weak and strong nuclear forces in terms of mathematical structures that arise in studying geometric symmetries. The predictions of Yang-Mills theory have been verified by countless experiments, and the theory is an important part of our understanding of how atoms are put together.

Despite that physical success, the theoretical mathematical underpinnings of the theory remain unclear. One particular problem of interest is the "mass gap," which requires that certain subatomic particles that are in some ways analogous to massless photons instead actually have a positive mass. The mass gap is an important part of why nuclear forces are extremely strong relative to electromagnetism and gravity, but have extremely short ranges.

The Millennium Prize problem, then, is to show a general mathematical theory behind the physical Yang-Mills theory, and to have a good mathematical explanation for the mass gap.

Read the Clay Mathematics Institute's official description of the Yang-Mills theory and mass gap problem here.

The Riemann Hypothesis

Going back to ancient times, the prime numbers — numbers divisible only by themselves and 1 — have been an object of fascination to mathematicians. On a fundamental level, the primes are the "building blocks" of the whole numbers, as any whole number can be uniquely broken down into a product of prime numbers.

Given the centrality of the prime numbers to mathematics, questions about how primes are distributed along the number line — that is, how far away prime numbers are from each other — are active areas of interest.

By the 19th century, mathematicians had discovered various formulas that give an approximate idea of the average distance between primes. What remains unknown, however, is how close to that average the true distribution of primes stays — that is, whether there are parts of the number line where there are "too many" or "too few" primes according to those average formulas.

The Riemann Hypothesis limits that possibility by establishing bounds on how far from average the distribution of prime numbers can stray. The hypothesis is equivalent to, and usually stated in terms of, whether or not the solutions to an equation based on a mathematical construct called the "Riemann zeta function" all lie along a particular line in the complex number plane. Indeed, the study of functions like the zeta function has become its own area of mathematical interest, making the Riemann Hypothesis and related problems all the more important.

Like several of the Millennium Prize problems, there is significant evidence suggesting that the Riemann Hypothesis is true, but a rigorous proof remains elusive. To date, computational methods have found that around 10 trillion solutions to the zeta function equation fall along the required line, with no counter-examples found.

Of course, from a mathematical perspective, 10 trillion examples of a hypothesis being true absolutely does not substitute for a full proof of that hypothesis, leaving the Riemann Hypothesis one of the open Millennium Prize problems.

Read the Clay Mathematics Institute's official description of the Riemann Hypothesis here.

The Birch and Swinnerton-Dyer conjecture

One of the oldest and broadest objects of mathematical study are the diophantine equations, or polynomial equations for which we want to find whole-number solutions. A classic example many might remember from high school geometry are the Pythagorean triples, or sets of three integers that satisfy the Pythagorean theorem x+ y= z.

In recent years, algebraists have particularly studied elliptic curves, which are defined by a particular type of diophantine equation. These curves have important applications in number theory and cryptography, and finding whole-number or rational solutions to them is a major area of study.

One of the most stunning mathematical developments of the last few decades was Andrew Wiles' proof of the classic Fermat's Last Theorem, stating that higher-power versions of Pythagorean triples don't exist. Wiles' proof of that theorem was a consequence of a broader development of the theory of elliptic curves.

The Birch and Swinnerton-Dyer conjecture provides an extra set of analytical tools in understanding the solutions to equations defined by elliptic curves.

Read the Clay Mathematics Institute's official description of the Birch and Swinnerton-Dyer conjecture here.

The Hodge conjecture

The mathematical discipline of algebraic geometry is, broadly speaking, the study of the higher-dimensional shapes that can be defined algebraically as the solution sets to algebraic equations.

As an extremely simple example, you may recall from high school algebra that the equation y = x results in a parabolic curve when the solutions to that equation are drawn out on a piece of graph paper. Algebraic geometry deals with the higher-dimensional analogues of that kind of curve when one considers systems of multiple equations, equations with more variables, and equations over the complex number plane, rather than the real numbers.

The 20th century saw a flourishing of sophisticated techniques to understand the curves, surfaces, and hyper-surfaces that are the subjects of algebraic geometry. The difficult-to-imagine shapes can be made more tractable through complicated computational tools.

The Hodge conjecture suggests that certain types of geometric structures have a particularly useful algebraic counterpart that can be used to better study and classify these shapes.

Read the Clay Mathematics Institute's official description of the Hodge conjecture here.