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Mathematics is awash with fiendish problems such as that of strong coupling, which crops up in physical systems from boiling water to superconductors where there are too many moving parts to be accurately modelled. Some researchers think they might now be on the way to solving that problem听(see 鈥�How magnets and boiling kettles encode the secrets of reality鈥�) 鈥� but what other thorns are there still in mathematicians鈥� sides?

In May 2000, the Clay Mathematics Institute in New Hampshire published a list of seven particularly intractable problems, and offered a million-dollar reward for the first correct solution to each. Only one of the Millennium Prizes has so far been claimed 鈥� for the Poincar茅 conjecture, which concerns a problem in four-dimensional geometry. The remaining six are up for grabs.

1. Navier鈥揝tokes existence and smoothness

The Navier-Stokes equations are used to describe the behaviour of fluids as they run out of a tap or flow over the wing of a commercial jet, which makes them incredibly important to solve. But their mathematical soundness is in question: for certain problems, it鈥檚 possible that the equations could malfunction to generate incorrect answers, or give no solutions at all. The existence and smoothness problem aims to sort out once and for all what鈥檚 really going on 鈥� and hopefully establish that the equations are a good fit to reality. Many mathematicians have tried 鈥� and failed 鈥� to find the answer, including recently Mukhtarbay Otelbaev of the Eurasian National University in Astana, Kazakhstan, who in 2014 claimed a solution, but later retracted it. Some physicists now think new advances in understanding strong coupling could now help crack the Navier-Stokes equations, too.

2. Riemann hypothesis

Prime numbers, those only divisible by themselves or one, are among the most important mathematical objects around. They crop up in nature and are a staple of cryptography, yet mathematicians still don鈥檛 know the pattern that describes how frequently they show up. The Riemann hypothesis suggests all primes lie in a valley that makes up part of a complex mathematical landscape known as the Riemann zeta function. Computers have checked this is true up to primes in their trillions, but a real proof must show the pattern holds for the infinity of all possible primes.

3. P vs NP

Why are some problems harder than others? Mathematicians like to classify puzzles according to the amount of effort you need to put into solving them. One set of puzzles, known as P, details problems that are relatively easy 鈥� nothing that would challenge an ordinary desktop computer. Another, NP, lists problems that might be hard, but whose solutions can easily be checked. The P vs NP question asks whether these two classes are in fact one and the same. Most mathematicians believe they aren鈥檛 but, ironically, are struggling to prove it.

4. Yang鈥揗ills mass gap

Yang-Mills theory provides a mathematical basis for our current understanding of elementary particle physics. Without it, we wouldn鈥檛 be able to say how many particles there are or what masses they should have. But there鈥檚 a problem. Experiments like the Large Hadron Collider at CERN near Geneva, Switzerland, as well as computer simulations suggest that there鈥檚 a minimum mass that particles can have 鈥� you can鈥檛 just conjure up a new one with arbitrarily low mass. But the distance between this mass and zero 鈥� the so-called 鈥渕ass gap鈥� 鈥� doesn鈥檛 appear to be contained within the framework of Yang-Mills theory. Solving the problem involves mathematically justifying the existence of this gap.

5. Birch and Swinnerton-Dyer conjecture

Equations known as elliptic curves describe wiggly shapes on a graph, and take the form y² = x³ + ax + b, where x and y are variables and a and b are fixed constants. They are used in cryptography and were essential in solving another long-standing and recently solved problem, Fermat鈥檚 Last Theorem. Mathematicians who work with these curves use another equation called the L-series to study their behaviour. The Birch and Swinnerton-Dyer conjecture says that if an elliptic curve has an infinite number of solutions, its L-series should equal 0 at certain points. Proving this is true would let mathematicians dive even deeper into these sorts of equations, although the practical applications are not immediately obvious.

6. Hodge conjecture

Mathematicians often find they can turn a problem in one field, such as algebra, into one from another field, such a geometry, to help them solve it 鈥� that鈥檚 what鈥檚 going on when you sketch an equation onto a piece of paper as a graph. Such graphs are two dimensional, meaning the related equations can only have two variables. So how do you use this trick on equations with three, four or even more variables? The answer lies in the field of algebraic geometry, which generalises this translation idea to higher dimensions.

Algebraic geometers work with techniques and concepts far more complex than simple equations and graphs, and have gradually figured out how to translate problems between them. The Hodge conjecture describes how you might do this for a particular type of mathematical object called a Hodge cycle 鈥� but until someone proves it right and claims the prize money, we鈥檒l never know for sure whether you can.