Chapter

From when we started to **study maths** in primary school: learning how to count and calculate, we have known about the fundamentals of mathematics.

In effect, for some people, maths – multiplication, fractions, and even statistics, allows them to better understand the world that we live in, studying it as a discipline and philosophy – not just to pass exams!

From secondary school through to sixth form, we are presented with a **series of mathematical concepts** which are well studied and *irrefutable*: there are concrete solutions to each concept and problem given – a solution you are often examined on! It may be easy to believe that the logic behind mathematics poses no further questions, that no further research is needed…

However, there are *certain problems in mathematics that have never been solved*, and even the greatest scientists and researchers haven’t been able to find their solutions.

These puzzles relate to our understanding of some of the most profound concepts in mathematics, and define or challenge our knowledge of basic math facts.

Perhaps you’ve undertaken your studies in maths only to better succeed in your academic career: passing exams, getting good coursework marks. However, could you be destined for greater things? You could be the first person to solve one of these mathematical mysteries!

Finding a solution to one of these seven problems could bag you $1 million! Interested now?

SuperProf is bringing you this list of unsolved mathematical problems, and we hope to one day read about you in the history books, having solved one (or more!) of them!

## Riemann Hypothesis

This problem is considered by many mathematicians to be one of the most difficult of all time. As a result, the Riemann hypothesis has never been solved!

This is surely the reason why today, so few researchers choose to study it: for fear of wasting their career on a mystery which seems impossible to solve.

David Hielbert had the Riemann hypothesis listed as number 8 on his list of problems presented at the Congress of Parisian Mathematicians in 1900. 100 years later, the Clay Mathematics Institute included it in its list of “Problems of the millennium”.

Solving the hypothesis would lead to a prize of $1 million!

Could this be a reason to take maths lessons, to maybe one day solve the problem known as the “holy grail of mathematicians”?

In 1859, Bernard Riemann published an article entitled “On the number of primes less than a given quantity”, without knowing that he had just posed the most complicated question in the history of mathematics.

His hypothesis began to tackle a question to which mathematicians haven’t been able to answer for the last 2000 years: the origin of prime numbers.

Following on from the works of his professor, Gauss, the German Riemann updated the Zeta function.

What does this mean? He constructed a 3d graph, and saw that the function only had its zeros at even negative numbers and complex numbers with a real portion of a ½. According to him, these zero points have a link with prime numbers.

Proving this link would help to discover the origin of the famous prime numbers.

If you would like to know more about those problems, search for "maths tutors near me" on Google.

## The Hodge Conjecture

Also appearing on the list of the seven problems of the millennium is the Hodge conjecture: uniting several mathematical skills that hadn’t previously been linked: algebraic topology and algebraic geometry.

According to this definition from the Clay Institute, the conjecture poses questions about the variety of complex projections (which are specific types of topological space) – Hodge objects are linear combinations with rational coefficients from classes associated with algebraic geometric objects.

Claire Voisin, a French mathematician, has worked on this hypothesis. According to her, proof would be a real mathematical treasure!

In an interview, she summarises that the Hodge Conjecture by explaining that a type of object, a variety of complex projections, are collections of points in a projected collection, defined by polynomial constraints.

Pretty complex, right?

Maybe this is the most difficult problem to solve, maybe not, but it’s certainly the most difficult to understand, thanks to the deep, in-depth knowledge of mathematics you must already possess in order to understand the puzzle to start with!

Solving it is a question of, amongst other things, geometry that we cannot visualize, making it even trickier to comprehend!

Perhaps some private maths tuition could help you get there?

## The Birch/Swinnerton-Dyer Conjecture

This particular conjecture is a question of algebraic equations – a math concept you're probably familiar with, having studied Algebra since secondary school!

Nether the less, you must have a certain skill in mathematics before trying to solve this particular conjecture! Perhaps a little knowledge of calculus could help you get started?

The conjecture attempts to define the number of distinct points on a elliptic curve.

It’s already pretty complicated to determine solutions to a polynomial equation (where x or y = 0), where x and y are both rational numbers…

This conjecture, also with a prize of $1 million, complicates things by suggesting that by suggesting that the solution depends on the number of solutions for every prime number P.

## The Navier-Stokes Equations

This one is a question of physics and fluid dynamics! Better put on your problem solving cap...

Less famous than Einstein’s E = MC^2, the Navier-Stoke equation has fascinated physicists and mathematicians alike, and is to do with the movement of fluids.

It consists of a non-linear differential equation, and its peculiarity is the fact that the equation is frequently used, even though we haven’t yet found its solution!

It’s used, amongst other things, to better understand the movement of currents in the oceans.

If you have some formidable mathematical or physics skills, proving the Navier-Stokes equation would give you the title of the 2^{nd} person to solve one of the seven Clay Institute problems, and walk away a millionaire!

Currently, only the Poincaré conjecture has been proven.

## The Yang Mills Equations

Another physics based problem, the Yang Mills theories aim to tackle problems in our understanding of the fundamental forces of the universe.

To explain these particles, Yang and Mills attempted to describe elemental particles by constructing a model based on geometric theories.

Their theory, which says that certain quantum particles have a positive mass, has been verified by a number of computer simulations.

Discovered by two physicists, the theory hasn’t been proven yet, and is still just an idea.

## P=NP

This puzzle is perhaps the most important of all.

Essentially, the resolution of this problem would solve many other problems, while for as long as it remains unsolved, so do many other problems in the fields of maths and computing. Many computations done today are known as NP-hard problems, because they fall into this category.

In P=NP, we call P the problem, where the solution is a group of elements from a given set.

Closely linked to the functioning of computers and algorithms, we could sum this problem up as the following question:

Can we determine, thanks to a calculation, what we can determine by luck?

Could you answer this as yet unanswered question?

Learn how to graph functions here.

## Ramsey Numbers

The Ramsey theorem is linked to order and to the models at the heart of various systems. According to this theory, true disorder cannot exist.

To summarise: if we draw n points on a sheet of paper, so that each point is linked to all the other points by either a red or blue line, n must be equal to 6 in order to be certain that there will be at least one triangle that is either red or blue.

Simply, we could ask what size our group must be for at least three of its members to be strangers, and three to have mutual connections. The answer to the problem is 6.

However, if we change the number 3 by 4, the problem is impossible to solve. Or at least, no mathematician up until today has succeeded.

Could you come up with the right formula?

## Lychrel Numbers and Palindromes

In order to understand the Lychrel numbers, you must first know the definition of a palindrome.

Palindromes can take the form of a number or words that, when read left-to-right, or right-to-left, read the same.

17371 is an example of a palindrome number, as it reads the same whether or not you start on the left or right.

When we repeatedly add a number with its inverse and the result doesn’t form a palindrome, it’s known as a Lychrel number.

59 isn’t a Lychrel number because…

59 + 95 = 154

154 + 451 = 605

605 + 506 = 1111

Effectively, we’ve ended up with another palindrome.

The smallest number for which we’ve not found a palindrome is 196, and this is exactly what impassions each mathematics researcher: not knowing exactly how to solve the problem... yet!

Even after more than 12 million repeated additions (thanks to the help of automation, of course!), we haven’t found a palindrome for the number 196!

Are you ready to pursue this kind of research?

Before trying to solve problems linked to algebra, geometry and physics, you must adopt a rigorous mathematical approach and immerse yourself in the scientific universe!

Throughout your school career, up to GCSEs, A levels, and degree level, you improve your memory and intellectual skills thanks to mathematics, and perhaps a home tutor could help you progress further?

Thanks to a private tutor’s personalised method of teaching, unique to you, you could improve your problem solving and analytical skills! And one day, perhaps you could solve one of these problems!