If you like math and you like to solve mysteries you will want to know how to solve math problems as soon as possible. But, not all problems can be easily solved. Here we have a few of them that are either impossible to solve or extremely hard so you will need time to deduce the outcome. Anyway, these problems are fun and they will keep you busy and interested in math. Who knows, maybe you can solve a problem that remained a mystery for decades or even centuries.
- Twin Prime
First of all, you need to know what prime numbers are. These are numbers (some even call them magical numbers) that are divisible by 1 and themselves only. There are countless of these numbers and experts are always looking for finding the new, highest one.
The main question here is a bit special. Are there pairs of prime numbers that differ only by 2? The best example is 41 and number 43. Keep in mind that as these numbers are increasing, finding the pairs is harder and harder. The theory on the other hand is simple. These should be infinite. The problem is, nobody proved this.
2. The Collatz
The Collatz conjecture is one of the best-known and at the first sight the simplest of all the problems. You can explain this problem to a child and he will understand it. You can even see this and many similar problems at a website that solves math problems on a daily basis but there is no solution for The Collatz conjecture. There are theories only. Let’s explain this problem.
You will start by picking a number. It can be any number. If it is an odd number, multiply it by the number 3 and then add 1. If it is an even number, divided it by 2. Once done you will have to repeat the steps but with the new number. An interesting fact is that you will end up with number 1, every single time! Try and you will see.
The main issue here is that this works with all numbers and there are no numbers that will provide a different result. Mathematicians have tried with millions of numbers and the result is always the same! It is possible that there is one number that defies this rule but nobody was able to find it. You can be the first one.
3. Riemann Hypothesis
Riemann Hypothesis is a very appealing problem. First of all, there is a $1 million reward for anyone who solves it. It has been offered for some time but nobody claimed it yet. Then we can see that the hypothesis is extremely simple and you won’t have a hard time understanding the problem.
This is Riemann zeta function and it is of impressive importance. Each s comes with an infinite sum. Then we can see an example in which s=2, then we have 𝜁(s) that is 1 + 1/4 + 1/9 + 1/16 + … This adds precisely to 𝜋²/6. The problem here is when s is not a simple number but rather complex. The hypothesis here is about when 𝜁(s)=0 which makes the problem extremely hard to solve. The function has special behavior that we can see throughout the special vertical line. The behavior will be present infinitely.
If mathematicians or you solve this hypothesis it would have a huge impact on the world and obviously the math. Sadly there are no insights that anybody will complete it in the near future. We can only hope that some of you will be the bright mind that solves this hypothesis.
4. The Beal
The Beal conjecture is another problem that looks simple and like something you can solve within minutes. But, it is one of the hard math problems with no apparent solution. Let’s start with this: Ax + By = Cz.
All of these (z, y, x, A, B, and C) are positive integers. This means that the numbers are bigger than 0. It also means that A, B, and number C all have a common prime factor. This means that each number should be divisible by the same number which is prime. In other words, numbers 5, 10, and 15 all have 5 as the prime factor which is 5.
The problem here is that nobody can solve the problem when z, x, and y are bigger than 2. The example 51 + 101 = 151 works. But the 52 + 102 ≠ 152 doesn’t. Can you solve the problem?
At the moment you can win $1 million dollars if you can solve it hence you may want to start working on it. Obviously, you will have to present the whole process and the proof that you solved this particular problem.
5. Goldbach’s conjecture
Goldbach’s conjecture may look like a simple problem that you can solve within seconds. It also looks like a variation of the Twin Prime problem we have covered. The problem is: will every number that is even and greater than 2 sum up two prime numbers?
At first sight, we would say yes and it is quite obvious. Here are the examples:
This is just a plain example. There are countless or better said the list is infinite. This was the initial of we can even say original design by Christian Goldbach who was a German mathematician. He developed this problem in 1742. Don’t forget that these days we don’t see 1 as the prime number. All integers that are positive and bigger than number 4 can be defined as the sum of specific primed, 2 of them.
Sadly there is no person who was able to prove this. An interesting fact is that there was a prize offered in the 2000s for solving this problem. Nobody got the prize hence you can deduce that the problem is either extremely hard or there is no solution.
6. Birch and Swinnerton-Dyer Conjecture
Birch and Swinnerton-Dyer Conjecture is big and important. First of all, it is one of 6 Millennium Prize Problems that are still a mystery. What this means is that it is one of the extremely hard math problems which can give you a prize if you solve one. All 6 are unsolved at the moment of writing this. This problem is not something we can explain in great detail and present you with all the data you can imagine. It is too complex and still unclear. We will try to do this in plain English.
Elliptic Curves is present here and the conjecture is based on this math topic. Fermat’s Last Theorem was solved with the help of Elliptic Curves who was used by Sir Andrew Wiles. In general, this is a powerful tool that can help you when nothing else will.
Sir Andrew Wiles is a function but a bit special one. The form can be something like y²=x³+ax+b. The main advantage here is that we can see special properties that cast specific insights into other topics such as Number Theory and Algebra.
Peter Swinnerton-Dyer and Bryan Birch who were both from the United Kingdom developed this problem in the 60s. The precise statement is very technical and also detailed. But, we know that over the years it had evolved so how it is a bit special and more detailed which can help you.
7. Kissing Number Problem
Kissing Number Problem comes with a nice and interesting name but this doesn’t mean that it is romantic and something you can solve within days. Many problems in math are called Sphere Packing Problems. There are a lot of variations and versions ranging from practical applications to pure math, something we all love. The main idea is the stacking of as many spheres as possible in one specific place. The best example would be fruits in the grocery bag. Some of these problems are relatively easy to solve and they have been solved already. Others are not solved ye. The best example here is the Kissing Number Problem.
If we stack a lot of spheres into the same spot we will have each one touching others. The kissing number is how many contacts one sphere makes with others or how many spheres one sphere is touching. If one sphere is touching 6 others, the kissing number is obviously 6.
Dimensions here have a huge role. Dimensions x and y are always used for coordination. An interesting fact is that if in Sci-Fi movies a character wants to travel to the second dimension, it doesn’t have any mathematical sense. You can’t go to the x-axis. Keep in mind that a one-dimensional thing is just a line. If we have two dimensions we will have a plane.
Mathematicians have solved the problem when the spheres are located in 1, 2, and 3 dimensions. In 1 dimension the solution is 2 or basically, your spheres can touch only two other spheres (one on each side). The solution for 3 dimensions comes from the 50s.
However, if we have more than 3 dimensions we don’t have a solution. The possibilities are not massive if you have up to 24 dimensions. This is still rare and not something most experts even know. But, when the numbers are much bigger there is no clue of any kind. Try and solve it.
The Final Word
All of these are hard math problems so don’t worry if you can’t solve one in a matter of minutes. In reality, these are problems that a mathematician will work on for decades or even the entire life. Yes, they sound simple but they are far from that. Some of these haven’t been solved for centuries. Luckily, no you can intrigue your brain and you can try to solve one. If you do, you can even win a monetary prize of up to $1 million. It is a worthy motivation.