The most unsolved problem in math is the Riemann Hypothesis, which concerns the distribution of prime numbers.
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The Riemann Hypothesis is arguably the most famous and important unsolved problem in mathematics. This hypothesis concerns the distribution of prime numbers, and was introduced by Bernhard Riemann in 1859 in his paper “On the Number of Primes Less Than a Given Magnitude.” The hypothesis suggests that the prime numbers are distributed in a non-random way, and that there is a connection between the distribution of primes and certain properties of the zeta function.
The zeta function of a complex variable s is defined as ζ(s) = ∑n=1 to ∞ 1/ns, where n ranges over all positive integers. The Riemann Hypothesis proposes that all non-trivial zeros of the zeta function, which lie in the critical strip 0<σ<1, have a real part equal to 1/2. In other words, the hypothesis claims that all of the non-trivial zeros of the zeta function lie on a vertical line in the complex plane.
The significance of the Riemann Hypothesis lies in its deep connections to other mathematical fields, such as number theory, mathematical physics, and cryptography. The Riemann Hypothesis has been described as “the most important unresolved problem in pure mathematics” by the Clay Mathematics Institute, which has offered a one-million-dollar prize for a proof of the hypothesis.
Despite the efforts of countless mathematicians over the past century and a half, the Riemann Hypothesis remains unsolved. As the mathematician Dan Rockmore has noted, “The Riemann Hypothesis has tantalized mathematicians for more than 150 years, and as far as anyone knows, it remains an unsolved problem.”
Some interesting facts about the Riemann Hypothesis include:
- The hypothesis has been verified to be true for the first 10 trillion non-trivial zeros of the zeta function.
- Some mathematicians believe that a proof of the Riemann Hypothesis could lead to breakthroughs in other fields, such as coding theory and quantum mechanics.
- The Riemann Hypothesis is one of the seven Millennium Prize Problems, a set of unsolved problems in mathematics that were identified by the Clay Mathematics Institute in 2000 and each of which carries a prize of one million dollars.
- The Riemann Hypothesis has inspired numerous books, articles, and documentaries, and has become a subject of cultural fascination. In the words of the mathematician Marcus du Sautoy, “The Riemann Hypothesis is the greatest unsolved problem in mathematics, inspiring books, plays, and films, and tantalising mathematicians with the promise of deep insight into the nature of prime numbers and the wider universe in which we live.”
In summary, the Riemann Hypothesis is a fundamental unsolved problem in mathematics that has captivated mathematicians and the public for over a century. Its resolution would have far-reaching implications for many fields of mathematics and science. As the mathematician Tim Gowers has said, “The Riemann Hypothesis is the most important open problem in mathematics not only because it is so rich in structure, but also because it is so accessible.”
Table: A list of some of the key properties of the Riemann zeta function, which is intimately connected to the Riemann Hypothesis.
| Property | Description |
|---|---|
| Analytic continuation | The Riemann zeta function can be extended to the whole complex plane. |
| Euler product | The function can also be expressed as an infinite product involving the primes. |
| Functional equation | The function satisfies a symmetry relation with respect to its argument. |
| Non-trivial zeros | The function has infinitely many zeros in the critical strip 0<σ<1, some of which are called non-trivial. |
| Prime number theorem | The distribution of prime numbers is related to the behavior of the zeta function. |
See a video about the subject
The “4 Weird Unsolved Mysteries of Math” video has presented four intriguing mathematical problems that have yet to be solved, starting with the Moving Sofa Problem, which focuses on finding the largest sofa that can be turned around a 90-degree corner without lifting it. The video also mentioned the Worm Problem or the Mother Worm’s Blanket, which involves finding the smallest blanket that can cover a sleeping baby worm in any position. Another problem is the shortest forest path, which aims to find the shortest path out of a specific shape of the forest, while the Magic Square of Squares problem is to find a functional 3×3 magic square made solely of square numbers. Despite the endless efforts of scientists and mathematicians alike, these challenges still remain unresolved, and many believe that they may never be solved in the future.
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One of the greatest unsolved mysteries in math is also very easy to write. Goldbach’s Conjecture is, “Every even number (greater than two) is the sum of two primes.” You check this in your head for small numbers: 18 is 13+5, and 42 is 23+19. Computers have checked the Conjecture for numbers up to some magnitude.
The Collatz Conjecture is the simplest math problem no one can solve — it is easy enough for almost anyone to understand but notoriously difficult to solve.
Despite many efforts, the Collatz conjecture has not yet been proven or disproven. It is considered one of the most famous unsolved problems in mathematics and has fascinated mathematicians for many years.
The Collatz conjecture is one of the most famous unsolved mathematical problems, because it’s so simple, you can explain it to a primary-school-aged kid, and they’ll probably be intrigued enough to try and find the answer for themselves. So here’s how it goes: pick a number, any number. If it’s even, divide it by 2.
The Riemann Hypothesis.
The Riemann hypothesis is a conjecture [ https://en.wikipedia.org/wiki/Conjecture ] that the Riemann zeta function [ https://en.wikipedia.org/wiki/Riemann_zeta_function ] has its zeros [ https://en.wikipedia.org/wiki/Root_of_a_function ] only at the negative even integers and complex numbers [ https://en.wikipedia.org/wiki/Complex_number ] with real part [ https://en.wikipedia.org/wiki/Real_part ] 1/2. Many consider it to be the most important unsolved problem in pure mathematics [ https://en.wikipedia.org/wiki/Pure_mathematics ] (Bombieri 2000 [ https://en.wikipedia.org/wiki/Riemann_hypothesis#CITEREFBombieri2000 ]). It is of great interest in number theory [ https://en.wikipedia.org/wiki/Number_theory ] because it implies results about the distribution of prime numbers [ https://en.wikipedia.org/wiki/Prime_numbers ]. It was proposed by Bernhard Riemann [ https://en.wikipedia.org/wiki/Bernhard_Riemann ] (1859 [ https://en.wikipedia.org/wiki/Riemann_hypothesis#…