# Best response to: what is the most difficult proof in mathematics?

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The most difficult proof in mathematics is currently unknown, as there are many unresolved conjectures and open problems.

## A more thorough response to your inquiry

Mathematics is a field that has been around for centuries and is constantly evolving as new ideas and theories are discovered. As such, it is difficult to pinpoint the most difficult proof in mathematics as there are many unresolved conjectures and open problems that remain unsolved to this day.

One example of an unresolved conjecture is the famous Riemann Hypothesis, which proposes a connection between the distribution of prime numbers and the zeros of the Riemann zeta function. According to the Clay Mathematics Institute, the Riemann Hypothesis is one of seven Millennium Prize Problems, with a \$1 million reward for anyone who can solve it.

Another important open problem is the Birch and Swinnerton-Dyer Conjecture, which concerns the behavior of rational points on elliptic curves. It has been listed as one of the seven Millennium Prize Problems as well.

In the words of mathematician Andrew Wiles, who famously proved Fermat’s Last Theorem after 358 years of failed attempts: “There’s a great thrill and satisfaction in knocking off a really hard problem – you complete it and you think, wow, that feels really good. But it’s much rarer than you think it is.”

It is worth noting that while these unsolved problems are certainly difficult, they are also exciting and inspire further research and exploration within the field of mathematics.

Here is a table listing some of the most famous unsolved problems in mathematics:

Problem Description
Riemann Hypothesis Connection between the distribution of prime numbers and zeros of the Riemann zeta function
Birch and Swinnerton-Dyer Conjecture Concerns the behavior of rational points on elliptic curves
Hodge Conjecture Concerns the topology of algebraic varieties
P vs. NP Problem Asks whether every problem that can be checked by a computer can also be solved efficiently by a computer
Navier-Stokes Equations Describes the motion of fluid substances, yet no complete proof of the equations has been found
Yang-Mills Existence and Mass Gap Seeks to better understand strong nuclear interactions
ABC Conjecture Involves connections between addition and prime numbers
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“There are no whole number solutions to the equation xn + yn = zn when n is greater than 2.” Otherwise known as “Fermat’s Last Theorem,” this equation was first posed by French mathematician Pierre de Fermat in 1637, and had stumped the world’s brightest minds for more than 300 years.

Despite being considered one of the most important unsolved problems in mathematics, the Riemann Hypothesis is yet to be proven or disproven. Many mathematicians have attempted to solve it, but the conjecture remains elusive.

Today’s mathematicians would probably agree that the Riemann Hypothesis is the most significant open problem in all of math. It’s one of the seven Millennium Prize Problems, with \$1 million reward for its solution.

Here is one of the hardest mathematical proofs of a problem that can be understood by a layman.

It is is called the “4-Color Problem”.

For most of human history maps were drawn in black or shades of black. When colors became widely available, they were used because it is easier to read a map that is colored. ‘Colored’ means coloring a map so that any two entities that share a border, use different colors. Think about a map of the states in America, or countries in Europe. Two states or countries that share a border must use different colors to be readable.

Around 1852, it was speculated that any such map could be colored with no more than 4 colors. No one could find a counter-example to this, but a proof eluded mathematicians.

Until 1976, that is. Then Appel and Haken, at the University of Illinois, used an IBM 360 that ran for weeks to prove the 4-Color Problem. It was the first significant proof that required a computer to prove because there were so many cases to consider that a…

The Principia Mathematica was written to derive a complete system of mathematics from pure logic; however, it is impossible to do so without any holes, leading to a few key assumptions that just feel right. Despite attempting to do a much broader and complicated thing, the book resulted in a 360-page proof that one plus one equals two. In a different context, the speaker suggests using a free daily newsletter called Morning Brew, which provides bite-sized, informative summaries of business, finance, and tech news, unlike social media feeds that waste time. The newsletter is free to sign up and only takes 15 seconds, making mornings much simpler.

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## More interesting questions on the topic

What does x3 y3 z3 k equal?
In mathematics, entirely by coincidence, there exists a polynomial equation for which the answer, 42, had similarly eluded mathematicians for decades. The equation x3+y3+z3=k is known as the sum of cubes problem.

What is the hardest thing to prove in math?
Today’s mathematicians would probably agree that the Riemann Hypothesis is the most significant open problem in all of math. It’s one of the seven Millennium Prize Problems, with \$1 million reward for its solution.

Herein, What is the world’s largest math problem? Mathematicians worldwide hold the Riemann Hypothesis of 1859 (posed by German mathematician Bernhard Riemann (1826-1866)) as the most important outstanding maths problem. The hypothesis states that all nontrivial roots of the Zeta function are of the form (1/2 + b I).

What are the 7 unsolved math problems? Clay “to increase and disseminate mathematical knowledge.” The seven problems, which were announced in 2000, are the Riemann hypothesis, P versus NP problem, Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier-Stokes equation, Yang-Mills theory, and Poincaré conjecture.

Besides, What are some brutally difficult math problems?
Here are some brutally difficult math problems that once seemed impossible to solve and some that still are. The Poincaré Conjecture, proposed by mathematician Henri Poincaré in 1904, is a problem that stumped the mathematics community for nearly 100 years.

What is the longest mathematical proof? Response: As of 2011, the longest mathematical proof, measured by number of published journal pages, is the classification of finite simple groups with well over 10000 pages. There are several proofs that would be far longer than this if the details of the computer calculations they depend on were published in full.

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People also ask, Are there proofs at every level of difficulty? As a response to this: There are proofs at every level of difficulty. You hear mostly about the most difficult ones because they’re, well, the most difficult. This is very vague. You could just as easily ask the opposite question: "why is it that so many interesting results have proofs that can be written out at lengths understandable by people?"

Also, Why are mathematical proofs so interesting?
Response: and the answer is: well, the entire reason they’re interesting is because they’re difficult. Mathematicians spend a ton of time exploring the space of possible proofs to find exactly those ones that are difficult and interesting.

Similarly, What is the longest mathematical proof? As of 2011, the longest mathematical proof, measured by number of published journal pages, is the classification of finite simple groups with well over 10000 pages. There are several proofs that would be far longer than this if the details of the computer calculations they depend on were published in full.

Also asked, Why is proof so difficult? One reason that university students find proof so difficult is that their experience with constructing proofs is typically limited to high school geometry [ Moore, 1994].

Keeping this in view, What is mathematical proof?
In reply to that: The expression "mathematical proof" is used by lay people to refer to using mathematical methods or arguing with mathematical objects, such as numbers, to demonstrate something about everyday life, or when data used in an argument is numerical.

Also question is, How many math problems are still unsolved?
Answer to this: In 2000, the Clay Mathematics Institute, a non-profit dedicated to “increasing and disseminating mathematical knowledge,” asked the world to solve seven math problems and offered \$1,000,000 to anybody who could crack even one. Today, they’re all still unsolved, except for the Poincaré conjecture.

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