The Most Difficult Mathematical Problems Are Not Solved; From The Riemann Hypothesis To P Vs. NP

The Most Difficult Mathematical Problems Are Not Solved; From The Riemann Hypothesis To P Vs. NP

In The World Of Mathematics, There Are Unsolved Problems That The Smartest Mathematicians Have Not Been Able To Solve For Years; These Questions Are So Important That A Million Dollar Prize have been Set Aside To Solve Some Of Them.

Let us be honest with ourselves from the beginning; Math is hard! So hard that Wikipedia’s list of unsolved problems in mathematics is staggeringly long. Some of these problems have remained unsolved for centuries, although the most intelligent people in the world have always been busy finding a way to solve them; For example, the issue of guessing odd integers was raised more than two thousand years ago and is considered one of the oldest unsolved mathematical problems.

Despite these difficulties, exciting events happen every moment in mathematics and physics, From hypergraphs and using them to solve a 50-year-old math problem to the great mysteries of physics and discovering an exact solution to a simple math problem that had remained unsolved for over 270 years.

This article has grouped some of the most important and famous unsolved math problems into three sections: the more complex issues, the seemingly simple, and the million-dollar difficulties, to give you a glimpse into this mysterious world.

The most complex mathematical problems have not been solved

What is the most difficult math problem in the world? The answer to this question is complicated; Because the level of “difficulty” is different for different people with other skills. Some mathematical problems, such as question 6 of the 1988 Mathematical Olympiad, are easy to understand; But it isn’t easy to solve.

This question was so complex that even the officials of the Olympiad could not solve it in less than six hours (it is interesting to know that Terence Tao, the winner of the 2006 Fields medal, scored only one point out of seven points for this question. Of course, he only scored 13 at that time. It was his year.) Some other questions also seem complicated, like the issue of Seven Bridges of Königsberg, But they have an easy solution.

Perhaps the best measure to measure the degree of difficulty of math problems is the number of people who have been able to solve them. Therefore, the most complex mathematical problems in the world are those that no mathematician has yet succeeded in solving.

Therefore, all six problems of the Millennium Prize, which you will get to know at the end of the article, and for solving each of them, a prize of one million dollars has been considered, are among the most complex mathematical problems.

In addition to the six, there are hundreds of other complex unsolved problems that, although not prize-winning, can be as influential in the development of mathematical sciences as the Millennium Prize problems. In the following, we will introduce some of them.

Separation problem (Separatrix Separation)

The hardest swinging pendulum math problem

A swinging pendulum can swing from side to side or rotate in a continuous circle. The point where the pendulum swings from one type of movement to the other is called the separatrix and can be calculated in most simple situations. But in the case where the pendulum moves at an almost constant speed, there is no longer any mathematical solution. Is there an equation that can describe this type of separatrix?

Brownian motion

Unsolved math problem - views and dimensions

Imagine a fragrance wafting through a room. The movement of each perfume molecule is random according to a process called Brownian motion, But the general way of gas movement is predictable.

In physics, Brownian motion is the random movement of particles immersed in liquid or gas due to the collision of these particles with atoms or molecules of the fluid. This phenomenon was named in honor of Robert Brown in 1827. However, its mathematical relationship was discovered by Albert Einstein in 1905, thus taking the final step in accepting the atomic theory of matter and the existence of atoms and molecules.

Brownian motion can be described with mathematical language thanks to Einstein’s theory, But this description is not 100% complete. According to this theory, if we want to reach the right solution, we have to break its rules; But if we try to stick to the rules, we will never get the exact answer. Is it possible to define another mathematical language for this phenomenon that both adhere to its rules and reach a clear solution? This is what the issue of views and dimensions seeks to solve.

Of course, the problem of Brownian motion has been partially solved. In 2000, Gregory Lawler, Oded Schramm, and Wendelin Werner proved that exact solutions to two Brownian motion problems could be found without violating its laws.

This proof earned the three scientists the Fields Medal, equivalent to the Nobel Prize. Stanislav Smirnov from the University of Geneva also managed to solve a problem related to the Brownian motion phenomenon, for which he also received the Fields Medal for this discovery.

Impossible theorems

Impossible math problems - Pi number

In the world of mathematics, many problems seem to have no solution. For example, consider pi, the ratio of a circle’s circumference to its diameter. When scientists proved that it is impossible to consider an end for the digits after the decimal point of pi and that this decimal point continues to infinity, it made a significant contribution to mathematics.

Likewise, physicists say it is impossible to find solutions to some problems, including precisely measuring the energy of electrons orbiting a helium atom. But can we prove this impossible?

Enclosed square problem

Difficult math problems - encircled square

Draw a closed curve on the paper. This curve can have as many bends and twists as you want. The only condition is that you must attach the beginning to the end, and it must not cut itself off. In the next step, try to find four points on the curve that can be used to draw a square. Can you do this for every angle?

This problem is known as the Inscribed Square Problem and asks whether it is possible to find four points to draw a square on any closed unbroken curve. This problem has been solved for several other geometric shapes, such as triangles and rectangles; For example, it has been proven that an infinity of circumscribed squares can be drawn in a circle and a court or that an obtuse triangle has precisely one circumscribed square; But whether it will work for the court or not is a bit vague, and so far no proof has been made by mathematicians.

Continuity hypothesis

The most difficult math problems - Continuity Hypothesis

Modern mathematics is full of infinities, from positive integers to infinite lines, triangles, spheres, cubes, polygons, etc. Modern mathematics has also proven that there are different values ​​of infinity. If the elements of a set can be put in a one-to-one correspondence with positive integers, we say that this set is a “countable infinity.” Therefore, the group of whole numbers and rational numbers are countable infinity.

In the 19th century, German mathematician Georg Cantor discovered that the set of real numbers was uncountable. This means that if we tried to assign a positive integer to every actual number, we would never be able to do so, Even if we use all integers. Consequently, its uncountable infinities can be considered “larger” than its countable infinities.

The Continuum Hypothesis in machine learning and unsolvable problems in mathematics asks whether it is possible to find a set of infinite numbers whose magnitude is strictly between countable and uncountable infinity. The continuum hypothesis is unsolved and has been proven intractable using current mathematical techniques. This means that although we do not know the truth of the continuum hypothesis, we do know that it cannot be proven true or false with current methods. Solving this hypothesis requires an entirely new framework that has yet to be developed.

Optimal chess strategy

Math Unsolved Questions - Optimal Chess Strategy

In game theory, “optimal strategy” is a limited set of steps that always leads to victory. Mathematicians have found optimal strategies for games like Doz, which, if you follow, you will always win the game.

For a long time, mathematicians have been looking for an optimal strategy for playing chess; that is, a particular set of dice moves guarantees a person’s victory in any situation. The problem of optimal chess strategy is interesting because even though we know there is a solution, we will probably never find it, and this is because of the enormous complexity of the chess game.

The reason for this complexity is that any program that wants to solve chess must be able to predict and compare all possible game variations to find the optimal move. This is while the number of possible games increases exponentially for every move made in chess.

Just take a look at the following table:

Number of moves Number of possible games
۱ ۲۰
۲ ۴۰۰
۳ ۸,۹۰۲
۴ ۱۹۷,۲۸۱
۵ ۴,۸۶۵,۶۰۹
۶ ۱۱۹,۰۶۰,۳۲۴

As the number of moves increases, the number of possible games increases incredibly quickly. After only five movements, the number of possible games reaches more than 69 trillion. It is estimated that the total number of possible positions on a chessboard is about 10 to the power of 120 (a number known as Shannon’s number).

This means that if a computer were to check all possible chess positions, it would take about 10 to the power of 90 years, roughly 8.3 x 10 to 79 times the current age of the universe, which is 13 billion years. Given these computational limitations, it seems unlikely that we will ever be able to solve chess, at least with current techniques.

Scientists indeed managed to develop artificial intelligence that can defeat even the great chess masters, But so far, none of them can solve the chess game itself. Instead, these models search through seas of terabytes of data to find game-winning strategies.

Unsolved math problems that look easy

Do not doubt that any mathematical problem that has not been solved so far is by no means simple; Solving these problems is either completely impossible or cannot be solved with current techniques.

However, some problems seem simple in the mathematical world, so simple that anyone with basic math knowledge can understand them. Still, these problems are so difficult to prove that no one has succeeded in solving them. In the following, you will get acquainted with a list of apparently simple math problems which are challenging to solve.

Guess the twin primes.

Simple unsolved math problem - Twin primes

Prime numbers are numbers that are divisible only by themselves. As far as we know, the number of prime numbers is infinite, and mathematicians are hard at work trying to find the following most significant prime number.

But there are some prime numbers whose subtraction is 2, such as 41 and 43. Is the number of these numbers also infinite? As the prime numbers get bigger, it becomes harder to find these twins (twin primes); But theoretically, these numbers should also be unlimited. The problem is that no one has yet been able to prove the infinity of dual prime numbers.

The problem with moving the sofa

Unsolved math problems - sofa moving problem

Most of us have probably faced the problem of moving the sofa and moving it through narrow corridors and corners of the wall when moving furniture to a new house. The question for mathematicians is this: What dimensions of the enormous sofa can you pass through the corner of the wall at a 90-degree angle without bending it, regardless of its shape?

Interestingly, the most significant volume that can fit in the corner of a 90-degree angle is called the “Sofa Constant.” No one knows exactly what this number is, But some sofas include in this angle and couches don’t work. Of course, mathematicians consider this sofa only in the door dimension and have nothing to do with its application in the real world.

Therefore, we know that this constant must be something between the dimensions of these two states. Currently, the only thing we know about this issue is that the Couch constant should be between 2.2195 and 2.8284.

Generally, we know that the sofa constant is between 2.2195 and 2.8284. But we have big sofas that we see this number is at least as significant as them. We also have sofas that don’t fit this size, so this size is smaller than that. Gly

Kulatz’s guess

Unsolved mathematical problems of Collatz Conjecture

The Collatz conjecture is one of the most famous unsolved problems in math, and because it looks so simple, you can explain it to elementary school kids, and they’ll probably like it enough to want to solve it.

Collatz Conjecture

Collatz conjecture function

The problem of Kulatz is as follows:

First, choose an arbitrary number. If this number is even, divide it by 2; if it is odd, multiply it by three and then add it by 1. Continue these steps for the new number obtained. The number you end up with will always be 1. For example, if the chosen number is 6, performing these steps will show these numbers: 6, 3, 10, 5, 16, 8, 4, 2, 1.

Example of Collatz Conjecture / Collatz Conjecture

Mathematicians have found millions of numbers that follow this rule, But the problem is that they have not been able to find a number that does not follow this rule. A considerable number that tends to infinity or a number stuck in a cycle may never reach one, But so far, no one has been able to find this number.

Bill’s guess

Beal Conjecture

Beal’s conjecture is another significant mathematical problem that seems simple, But no one has solved it yet.

According to this problem, if Ax + By = Cz and A, B, C, x, y, and z are all positive integers (numbers greater than zero), A, B and C must all have a common prime factor. The common prime factor means that each number must be divisible by the same prime number. For example, the common prime factor of the numbers 15, 10, and 5 is equal to 5, all divisible by the prime number 5.

This problem seems simple, and you may have solved it in your high school algebra class. But the problem is that mathematicians have not yet been able to solve Bill’s conjecture with x, y, and z more significant than 2. For example, if our first common factor is 5, 51 + 101 = 151, but 52 + 102 ≠ 152.

A billionaire raised this problem from Texas named Andrew Beale. The person who finally succeeds in solving it will receive a one million dollar prize from the American Mathematical Society.

Goldbach’s conjecture

The hardest math questions - Golbach's conjecture

Goldbach’s conjecture, like the twin prime numbers conjecture, is another unsolved problem about prime numbers that are seemingly simple but extremely difficult to solve. This problem asks whether every even integer greater than 2 is the sum of the first two numbers. You may say that it is clear that the answer is positive because the number 4 is the sum of the first two numbers, 3 and 1, or the number 6 is the sum of the first two numbers, 5 and 1, and this process continues in the same way.

This problem was solved similarly by the German mathematician Christian Goldbach, who proposed it in 1742. He confidently said, “any even integer greater than four can be written as the sum of the first two numbers.”

But despite centuries of efforts, no one has been able to prove that this rule always works for all numbers. The truth is that if we keep making the numbers more prominent and more significant and keep going, we might end up with a number that isn’t equal to the sum of the first two numbers, or we might end up with a number that violates all the rules and logic we’ve used so far. Undoubtedly, mathematicians will not stop trying until they find an answer to this problem.

To date, Goldbach’s conjecture was verified for all even integers up to 4 in 1018; But its analytical proof is still far from the reach of mathematicians. However, the consensus is that this conjecture is correct due to the nature of the distribution of prime numbers. Because the larger an integer, the more likely it can be expressed as the sum of two other numbers. Therefore, the larger an integer, the more likely that at least one of these combinations consists of only prime numbers.

Prize-winning unsolved math problems (Millennium Prize Problems)

The Millennium Prize Problems are seven mathematical problems presented by the Clay Mathematics Institute in 2000 to celebrate the new millennium. Whoever can solve one of these problems will win a $1 million cash prize, and solving these problems will have significant impacts on the field or even beyond.

These seven issues are:

  • P vs. NP
  • Hodge’s guess
  • Riemann hypothesis
  • Young-Mills theory
  • Navier-Stokes equations
  • Birch and Swinnerton-Dyer conjecture
  • Poincaré conjecture

Among these seven problems, Poincaré’s conjecture was solved in 2003 by Grigori Perelman, a Russian mathematician; However, he refused to accept the General Society Award and, of course, all other awards and medals for his achievements.

More than two decades have passed since the issues of the Millennium Prize were raised, and six other problems remain unresolved. We will continue to explain these issues; Maybe you can solve them!

Riemann hypothesis

Riemann Hypothesis

The most important unsolved problem in pure mathematics is known as the Riemann Hypothesis. This issue was raised by Bernhard Riemann, a German mathematician of the 19th century. His works in analysis and differential geometry became the mathematical basis of the theory of general relativity.

The Riemann hypothesis has remained unsolved since 1859. It is so complex that David Hilbert, one of the most influential mathematicians in the creation and development of quantum mechanics and relativity theory, said about it:

If I were to wake up after a thousand years, the first question I would ask would be: Has the Riemann hypothesis been proven?

It is interesting to know that in 1900, Hilbert raised twenty-three mathematical questions that had not been solved until then, and Riemann’s hypothesis was one of them. Some of these questions, known as Hilbert’s problems, were translated and significantly impacted 20th-century mathematics.

The Riemann hypothesis asks you to prove under what conditions the Riemann zeta function equals zero. In other words:

Riemann zeta function / reimann zeta function

This function seems simple on the surface, But its complexity appears on the graph. For example, to the chart |ζ(1/2+iy)| (vertical axis) as a function of y (horizontal axis).

As you can see, the zeta function approaches zero for values ​​14, 21, 25, and so on on the horizontal axis. These are called zeta function zeros, and they are essential because their behavior is exciting. Riemann’s hypothesis is a proposition about the distribution of these zeros. Riemann says that the zeta function only goes to zero when dealing with negative even integers and complex numbers with a natural part of 1/2. The problem is that although more than 250 million zeros have proven this hypothesis, it has not yet been proven that this applies to all zeros.

Riemann hypothesis - distribution of prime numbers

Distribution of prime numbers (yellow lines) in the set of natural numbers from 0 to 1000

The Riemann hypothesis is critical because prime numbers (divisible only by one and themselves) are mathematics’s most fundamental and mysterious concept. When we write the prime numbers as a linear series, no pattern appears in how they are distributed; therefore, we cannot predict all the prime numbers.

But when we plot these numbers with the help of the Riemann zeta function, an interesting pattern of Riemann zeros appears on it. If we can fix it for all numbers, we can say that we have finally discovered the hidden way of distributing prime numbers. Thus, with very high accuracy, we can determine the number of prime numbers in any given interval.

You may ask, what the importance of having a function to define prime numbers is? Many mathematicians see prime numbers as the building blocks of all other numbers because you can get to any number using prime numbers.

In the Riemann hypothesis, the range that is created on the number line of values ​​that make the zeta function zero is like the distances between energy levels in quantum systems, and this means that there is a relationship between the components of numbers with prime numbers and the details of matter with atoms. Solving this hypothesis will lead us to a new understanding of matter.

The P vs. NP problem

Prize-winning math problems - P vs. NP

P vs. NP is a significant unsolved problem in computer science and asks whether any problem whose answers can be evaluated quickly (NP) can also be solved promptly (P). Computer scientist Steven Cook raised this issue in 1971.

Let’s take an example to understand this issue better. If they give you a number and say that this number is obtained from the product of the first two numbers, can you reach the correct answer?

If this number is small, the answer is simple. For example, 15 is obtained by multiplying two numbers, 5 and 3. But if the desired number has 200 digits, it will take years to find its double.

Now let’s reverse this question; If they give you the first two numbers and say whether the product of these two results in the number x, finding the answer to this question is as easy as doing the multiplication operation.

In other words, you can quickly evaluate the accuracy of the answer by multiplying these two numbers. But as you have seen, on the contrary, this case takes so much time that it is almost impossible to solve it.

In computer science, a problem whose answers can be determined quickly is called P, and a problem whose solutions can only be rapidly verified is called NP.

It is essential that the problems can be solved quickly, or in the language of computer science, the time to execute their algorithm is “Polynomial Time”; if solving a problem takes hundreds or thousands of years, it is practically impossible to solve it.

Steven Cook’s question asks precisely this:

Can we have a polynomial-time algorithm for P for every polynomial-time NP algorithm?

The day someone can finally prove P=NP, many mathematicians will be out of a job because P=NP means that proving a mathematical theory is the same as evaluating the correctness of its answers.

Even worse, all banking systems also fail; Because the decryption of encrypted passwords with a vast multiple of prime numbers is possible in a fraction of a second. To get more familiar with this topic, I suggest the article Shor Algorithm in simple language; Read Data Decryption in Quantum Computer.

Hodge’s guess

The hardest math problems - Hodge Conjecture

Hodge’s conjecture is one of the significant unsolved problems in algebraic geometry and mixed geometry, which examines how to form complex mathematical structures from simple components and tries to connect these two different mathematical concepts.

In the 20th century, mathematicians discovered a critical way to observe and study complex objects by stacking increasingly large objects together to get the closest shape to the original.

This technique was so valuable that it was used in many other fields, and eventually, the complex objects that mathematicians classified in this way were used in amazing inventions.

Unfortunately, through these generalizations, the geometric origin of this process was lost, and the attempt was to link these components together without a geometric formula or support. Now Hodge’s conjecture asks if this concept has a geometric interface.

Young-Mills theory

Unsolved mathematical problem - Young-Mills theory

The Yang-Mills Theory is another prize-winning unsolved problem in the field of quantum physics. This theory defines particles using mathematical symmetry.

During the past six decades, the Young-Mills theory has become a cornerstone of theoretical physics; Because only the many-body quantum theory of relativity seems to be completely compatible with the four dimensions of spacetime, and for that reason, it is the basis of the standard model of particle physics, which has been proven to be the correct theory for the energies we can measure.

The Young-Mills theory is a generalization of the unified theory of electromagnetics, or “Maxwell’s equations,” proposed by James Clerk Maxwell, a Scottish physicist. It describes subatomic particles’ weak force and vital force in terms of geometric structure or quantum field.

This theory was proposed in 1954 by two physicists, Chen Ningyang and Robert L. Mills. It relies on the property of quantum mechanics called “Mass Gap,” which is the energy difference between the lowest level (vacuum) and the next lowest level and is equivalent to the mass of the lightest particle. Scientists believe that the mass gap is the factor that caused the vital force to exist only at minimal distances, that is, inside atomic nuclei.

Young Mills’ theory describes the unity of the electromagnetic force and the weak point; The first force causes the electrons to revolve around the proton, and the second force causes a neutron to split into an electron and a proton.

The difference between these two forces is like the difference between a moon that rotates while orbiting the planet and a moon that does not rotate while orbiting the earth. The force that keeps the moon in orbit is the same regardless of whether it is spinning. This is what integration means; To show that there is the same force behind these two different things.

Navier-Stokes equations

Navier-Stokes Equations / Navier Stokes Equations

Navier-Stokes Equations (Navier Stokes Equations) is another Millennium Prize problem related to differential equations describing compressible fluids’ motion. Briefly, the Navier-Stokes equations describe the behavior of fluids.

This equation is obtained by applying Newton’s second law to fluids. The flight of airplanes, electricity generation, weather forecasting, and even the construction of boats and ships depend on it. Even the Pixar animation company uses Navier-Stokes equations to animate their works.

Although these equations seem simple, they get complicated quickly in 3D mode. Princeton University professor Charles Fefferman says: “You can start solving Navier-Stokes equations relatively easily and with high confidence, But the solutions can be incredibly unpredictable.”

If mathematicians can take the Navier-Stokes phenomenon out of this unpredictable state, dramatic changes in fluid dynamics will be achieved. According to Fefferman, if these equations are proven, “it will be a tremendous achievement at the highest level.”

Brash and Swinnerton-Dyer conjecture

The hardest math problems - Bresch and Swinnerton-Dyer conjectures

In the early 1960s in England, British mathematicians Brian Brash and Peter Swinnerton-Dyer used the EDSAC computer, one of the first computers made in England, to conduct numerical research on elliptic curves. Based on these numerical results, they proposed the Birch and Swinnerton-Dyer conjecture, which is the last unsolved $1 million problem in this list.

Bresch and Swinnerton-Dyer’s conjecture says that an elliptic curve has an infinite number of rational points (solutions) if the corresponding function is equal to zero and a finite number of reasonable points if the process is not zero. In other words, this problem wants to prove that if an elliptic curve has an infinite solution, it will be zero at specific points of the series L.

This theory is widely used in cryptography and is very important for solving many problems, including Fermat’s final theorem.


All the issues mentioned in this article, especially the issues of the Millennium Prize, have not been raised for ordinary people, and it is difficult even to explain and understand them, let alone solve them. Perhaps the solution to many of them will require techniques that humans will achieve centuries later. By focusing on these issues, mathematicians are trying to open a path toward the future.

For this reason, solving these problems does not take time; every time a scientist can solve even one of these complex problems, scientific progress accelerates. These mathematical problems will lead to the formation of new theories, and their value lies in this.

In mathematics, especially in the field of number theory, where the problem is easy, But solving it isn’t easy, patience is more important than anything. Progress is made little by little; if we keep digging, we will eventually reach the diamond. Mathematicians say you should consider these issues for a long time and give them importance.

Frequently asked questions

What is the most critical unsolved math problem?

The most important unsolved problem in pure mathematics is known as the Riemann Hypothesis. This issue was raised by Bernhard Riemann, a German mathematician of the 19th century, and he wants you to prove under what conditions Riemann’s zeta function is equal to zero.

This problem tries to find a specific pattern for the distribution of prime numbers, which are mathematics’s most fundamental and mysterious concepts.

What are the issues of the Millennium Prize?

The Millennium Problems are seven mathematical problems, the solution of each of which brings a prize of one million dollars. These problems are P versus NP, Hodge’s conjecture, Riemann hypothesis, Young-Mills theory, Navier-Stokes equations, Birch and Swinnerton-Dyer conjecture, and Poincaré’s belief, the last of which was solved in 2003 by Grigory Perelman, a Russian mathematician.

What is the P vs. NP issue?

P versus NP is a significant unsolved problem in computer science and one of the Millennium Prize problems. It asks whether any problem whose answers can be evaluated quickly (NP) can also be solved promptly (P).

The day someone can finally prove P=NP, many mathematicians will be out of a job because P=NP means that proving a mathematical theory is the same as evaluating the correctness of its answers.