![]() Collatz Conjectureįirst, pick any positive number n. Solving the continuum hypothesis would require a new framework for set theory, one which has not been created yet. This means that, while we do not know the truth of the continuum hypothesis, we know that it can neither be proven nor disproven using the resources of modern set theory. The continuum hypothesis is a bit different than other problems on this list because, not only has it not been solved, it has been proven to be unsolvable, or at least, unsolvable using current mathematical techniques. The continuum hypothesis asks whether or not there exists a set of numbers that is an infinity whose magnitude is strictly between countably and uncountably infinite. Thus, uncountable infinities can be considered “bigger” than countable infinities. This means that if we tried to go through and assign a positive whole number to every real number, we would never be able to do it, even if we used all the whole numbers. In the 19th century, Georg Cantor discovered that the set of real numbers is uncountable. So the set of whole numbers is a countable infinite and so is the set of all rational numbers. We say a set of elements is countably infinite if the elements of that set can be put into a 1-to-1 correspondence with the positive whole numbers. Modern math has also proven that there are different magnitudes of infinity as well. There are infinite positive whole numbers (1,2,3,4…) and an infinite amount of lines, triangles, spheres, cubes, polygons, and so on. Modern math has infinities all over the place. The theorem has not been proven for the general case of any closed curve though. For example, it has been proven that circles and squares have an infinite amount of inscribable squares, obtuse triangles have exactly one, while right and acute triangles have exactly 2 and 3 respectively. The inscribed square theorem has been proven for a number of special cases of curves. The inscribed square problem asks whether every possible closed non-intersecting curve contains the 4 points of a square. This is known as the inscribed square problem. Credit: C Rocchini via WikiCommons CC-BY SA 3.0 The inscribed square problem concerns whether or not any generic closed non-intersecting curve contains the 4 points of a square. Next, try to find some 4 points located on the curve such that you can draw a square using those points. The curve can have as many squiggles and bends as you want the only conditions are that you have to close it end-to-end and it cannot intersect itself. Therefore, the larger an integer is, the more likely that at least one of these combinations will consist of only primes. In general, the larger an integer is, the more likely it can be expressed as the sum of two numbers. ![]() The informal justification for this claim comes from the nature of the distribution of prime numbers. Although mathematicians do not have a rigorous proof yet, the general consensus is that the conjecture is true. To date, the Goldbach conjecture has been verified for all even integers up to 4 × 10 18 but an analytic proof still eludes mathematician. The first 50 even numbers written as the sum of two primes. The Goldbach conjecture was first proposed by German mathematician Christian Goldbach in 1742, who posited the conjecture in correspondence with Leonhard Euler. GB: “Every even integer greater than 4 can be written as the sum of two prime numbers.” The question is, can you keep doing this forever? That is, can you write every possible even natural number as the sum of two primes? The Goldbach conjecture answers this question in the affirmative. For our first 5 elements of our list, we get: Next, take each even number and try to rewrite it as the sum of 2 prime numbers. First, take all the even natural numbers greater than 2 (e.g. Let’s start our list with an extremely famous and easy-to-understand problem. With that in mind, we are going to take a look at 6 of the most difficult unsolved math problems in the world. Therefore, it stands to reason that the hardest math problems in the world are ones that no mathematician has solved yet. Others such as the 7 Bridges of Königsberg problem seem complex but have a deceptively simple answer.Ī reasonable metric to determine how “difficult” a math problem is could be the number of people that have solved it. Some math problems, such as the infamous question 6 of the 1988 Math Olympiad are easy to understand but monstrously complex to solve. “Difficulty” is a subjective metric and what is difficult for some may not be difficult for others. What is the hardest math problem in the world? The answer to that question is tricky.
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