# Immediate reaction to: what math problems were once impossible to solve?

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Some math problems that were once impossible to solve include squaring the circle, trisecting an angle, and doubling the cube using only a compass and straightedge.

For centuries, many mathematicians have dedicated their lives to solving problems that were thought to be impossible. Among these problems were three famous ones that could only be solved using a compass and straightedge: squaring the circle, trisecting an angle, and doubling the cube.

Squaring the circle involves constructing a square with the same area as a given circle using only a compass and straightedge. It was proved in 1882 that this problem cannot be solved using only these tools. Trisecting an angle involves dividing a given angle into three equal parts using only a compass and straightedge. This problem was also proved to be impossible using only these tools. Doubling the cube involves constructing a cube with twice the volume of a given cube using only a compass and straightedge. Again, this problem was proved to be impossible using only these tools.

These problems all fall under the category of “constructible geometry,” which is based on the idea of using only a compass and straightedge to create geometric constructions. Although the three problems mentioned above cannot be solved using only these tools, there are many other problems that can be solved using them.

In the words of mathematician Hermann Weyl, “The question ‘What is mathematics?’ was answered by Gauss as follows: ‘Mathematics is the queen of the sciences and number theory is the queen of mathematics.’ ” Number theory, which deals with the properties of numbers, is just one branch of mathematics that has fascinated scholars for thousands of years.

Some interesting facts about the field of mathematics include:

• The concept of zero was invented independently by both the Mayans and the Indians.
• The ancient Greeks were obsessed with the idea of finding geometric solutions to problems that could only be solved algebraically.
• It was once believed that there were only five “platonic solids” (regular polyhedra), but it was later discovered that there are actually thirteen.
• Mathematicians are still working to solve the Riemann Hypothesis, a problem involving the distribution of prime numbers, that was first proposed in the mid-1800s.

Table:

Problem Description Cannot be solved with compass and straightedge?
Squaring the circle Constructing a square with the same area as a given circle Yes
Trisecting an angle Dividing a given angle into three equal parts Yes
Doubling the cube Constructing a cube with twice the volume of a given cube Yes
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Overall, even though there are some math problems that were once impossible to solve, the history and ongoing discoveries in mathematics continue to inspire and fascinate people around the world. As the great mathematician Pierre-Simon Laplace once said, “It is therefore from the study of the mathematical sciences, and especially from geometry, that taste is acquired for the beauty of the intellectual arts.”

The Collatz Conjecture is a problem in mathematics that is said to be incredibly difficult to solve. The problem involves determining whether or not a set of positive integers will eventually end up in a loop created by applying two rules. Professional mathematicians have been unable to solve the problem, but Jeffrey Lagarias is the world authority on the conjecture.

Unsolved Problems

• The Goldbach conjecture.
• The Riemann hypothesis.
• The conjecture that there exists a Hadamard matrix for every positive multiple of 4.
• The twin prime conjecture (i.e., the conjecture that there are an infinite number of twin primes).
• Determination of whether NP-problems are actually P-problems.

There are many unsolved problems in mathematics. Some prominent outstanding unsolved problems (as well as some which are not necessarily so well known) include 1. The Goldbach conjecture. 2. The Riemann hypothesis. 3. The conjecture that there exists a Hadamard matrix for every positive multiple of 4.

Of the original seven Millennium Prize Problems listed by the Clay Mathematics Institute in 2000, six remain unsolved to date: [6]

• Birch and Swinnerton-Dyer conjecture
• Hodge conjecture
• Navier–Stokes existence and smoothness

The remaining six unsolved problems are the Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier–Stokes existence and smoothness, P versus NP problem, Riemann hypothesis, and Yang–Mills existence and mass gap.

The proof that demonstrates the impossibility of trisecting an angle uses Galois theory. Galois theory can also be used to show that certain polygons cannot be constructed with compass and straightedge, and was originally used to show that, in general, polynomials of degree ≥5 are not solvable.

To be specific, an n-gon is constructible via compass and straightedge ⟺n=2km∏r=1pr, for k,m∈Z≥0 and the pr’s distinct Fermat primes.

Without getting into too much detail, Galois theory is a subset of abstract algebra which links together concepts in group theory and field theory, beginning with the observation that the set of automorphisms of a field forms a group. At any rate, I’m unsure of your level of background, so I’ll just post a few Wikipedia links for to wet your appetite if you’re interested:
• Galois theory
• Groups
• Fields
• Automorphisms

Of course, there are many, many answers to your question, so I’ll leave my post at that and let others have the opportunity to post more.

You will most likely be interested in these things as well

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In respect to this, Are there math problems that haven’t been solved? As a response to this: Of the original seven Millennium Prize Problems listed by the Clay Mathematics Institute in 2000, six remain unsolved to date: Birch and Swinnerton-Dyer conjecture. Hodge conjecture. Navier–Stokes existence and smoothness.
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Consequently, What mathematical equation has never been solved?
Answer: 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. So what is the Collatz Conjecture and what makes it so difficult?

What are the 7 unsolved maths 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.

Also to know is, What is the hardest math problem that has not been solved?
Goldbach’s Conjecture
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.

Are there any mathematical problems that have never been solved? The answer is: So Far this has never been solved. As you can see in the equations above, there are several seemingly simple mathematical equations and theories that have never been put to rest. Decades are passing while these problems remain unsolved. If you’re looking for a brain teaser, finding the solutions to these problems will give you a run for your money.

Subsequently, Are there any unsolved problems in Algebra and model theory?
In reply to that: The Erlagol Notebook ( Russian: Эрлагольская тетрадь) lists unsolved problems in algebra and model theory. Of the original seven Millennium Prize Problems listed by the Clay Mathematics Institute in 2000, six remain unsolved to date:

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Considering this, Are there simple mathematical equations that have never been put to rest? As you can see in the equations above, there are several seemingly simple mathematical equations and theories that have never been put to rest. Decades are passing while these problems remain unsolved. If you’re looking for a brain teaser, finding the solutions to these problems will give you a run for your money. See the 11 Comments below.

Beside this, Is Maths really hard?
Response: We all know that maths is really hard. So hard, in fact, that there’s literally a whole Wikipedia page dedicated to unsolved mathematical problems, despite some of the greatest minds in the world working on them around the clock.

Correspondingly, Are there any mathematical problems that have never been solved? The response is: So Far this has never been solved. As you can see in the equations above, there are several seemingly simple mathematical equations and theories that have never been put to rest. Decades are passing while these problems remain unsolved. If you’re looking for a brain teaser, finding the solutions to these problems will give you a run for your money.

Consequently, Are there any unsolved problems in Algebra and model theory?
Response will be: The Erlagol Notebook ( Russian: Эрлагольская тетрадь) lists unsolved problems in algebra and model theory. Of the original seven Millennium Prize Problems listed by the Clay Mathematics Institute in 2000, six remain unsolved to date:

Then, Are there simple mathematical equations that have never been put to rest?
The answer is: As you can see in the equations above, there are several seemingly simple mathematical equations and theories that have never been put to rest. Decades are passing while these problems remain unsolved. If you’re looking for a brain teaser, finding the solutions to these problems will give you a run for your money. See the 11 Comments below.

In respect to this, Can you solve the world’s hardest equations? Now He’s Solving the World’s Hardest Equations. 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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