The most extensive proof in mathematics is the classification of finite simple groups, which took several decades and involved the collaboration of hundreds of mathematicians.

## Response to the query in detail

The classification of finite simple groups is widely recognized as the most extensive proof in mathematics, taking nearly two centuries to complete. The theorem asserts that every finite simple group is one of a specific set of groups, which are themselves incredibly complex.

This monumental effort involved the collaboration of hundreds of mathematicians from around the world, and required the development of entirely new mathematical techniques. The proof was finally completed in 2004, with the publication of a series of papers totaling more than 15,000 pages.

In the words of John Conway, one of the mathematicians involved in the classification effort, “It is difficult to overstate the importance of the classification of finite simple groups … It represents the culmination of a subject that has occupied many of the best mathematical minds for over a century.”

Here are some interesting facts about the classification of finite simple groups:

- The first significant progress towards the classification was made in the late 19th century by mathematicians such as Camille Jordan and Ludwig Sylow.
- The classification was largely motivated by the desire to understand the structure of Lie groups, which are important in physics and geometry.
- The sheer scale of the classification effort meant that it was divided into several separate cases, each with its own set of proofs and techniques.
- The proof involved many different areas of mathematics, including algebra, geometry, combinatorics, and analysis.
- One of the most famous cases in the classification is the so-called “Monster group,” which has over 10^53 elements.
- The proof of the classification required the development of entirely new mathematical structures, such as “quasithin groups” and “quasicentralizers.”
- The classification has had many important consequences in mathematics and physics, including the discovery of new types of symmetry and the development of the theory of “moonshine.”

Here is a table showing the different types of finite simple groups, as classified by the theorem:

Type of Group | Approximate Number of Elements |
---|---|

Cyclic | Unlimited |

Alternating | Finite |

Lie type | Finite |

Sporadic | Finite |

Tits | Finite |

Others | Finite |

In conclusion, the classification of finite simple groups is a truly magnificent achievement in mathematics, representing the culmination of centuries of work by some of the greatest minds in the field. The depth and complexity of the theorem, as well as its many implications for other areas of mathematics and science, ensure that it will remain a subject of study and fascination for generations to come.

## See related video

The Langlands Program is a grand unified theory of mathematics that connects two disparate continents of mathematics – number theory and harmonic analysis. It was developed by Robert Langlands in the 1960s and has since been used to solve difficult problems in both fields. Two mathematicians, Srinivasa Ramanujan and Pierre Delign, are credited with making significant contributions to the theory.

## See further online responses

The puzzle that required the 200-terabyte proof, called the Boolean Pythagorean triples problem, has eluded mathematicians for decades. In the 1980s, Graham offered a prize of US$100 for anyone who could solve it.

Classification of finite simple groupsAs 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.

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.

I will illustrate with one of my favorite problems.

Problem: There are 100 very small ants at distinct locations on a 1 dimensional meter stick. Each one walks towards one end of the stick, independently chosen, at 1 cm/s. If two ants bump into each other, both immediately reverse direction and start walking the other way at the same speed. If an ant reaches the end of the meter stick, it falls off. Prove that all the ants will always eventually fall off the stick.

Now the solutions. When I show this problem to other students, pretty much all of them come up with some form of the first one fairly quickly.

Solution 1: If the left-most ant is facing left, it will clearly fall off the left end. Otherwise, it will either fall off the right end or bounce off an ant in the middle and then fall off the left end. So now we have shown at least one ant falls off. But by the same reasoning another ant will fall off, and another, and so on, until they all fall off.

Solution 2: Use symmetry: Im…

## I am sure you will be interested in this

Accordingly, **What is the biggest proof in math?** The proof, which concerns the classification of mathematical symmetry groups – a concept aptly known as the "Enormous Theorem" – **took 100 mathematicians three decades and some 15,000 pages of workings to pin down**.

**What is the most complex mathematical proof?**

In reply to that: 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.

**What is the hardest equation ever solved?** Response: x3+y3+z3=k, with k being all the numbers from one to 100, is a Diophantine equation that’s sometimes known as "summing of three cubes."

Subsequently, **What are the 3 types of proofs?** There are many different ways to go about proving something, we’ll discuss 3 methods: **direct proof, proof by contradiction, proof by induction**. We’ll talk about what each of these proofs are, when and how they’re used.

Also, **What is the longest mathematical proof?**

Answer will be: This is a list of unusually long mathematical proofs. Such proofs often use computational proof methods and may be considered non-surveyable . 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.

In this way, **What is mathematical proof?**

Answer: 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.

In this manner, **Are mathematical proofs analytic or synthetic?**

A classic question in philosophy asks whether mathematical proofs are analytic or synthetic. Kant, who introduced the analytic–synthetic distinction, believed mathematical proofs are synthetic, whereas Quine argued in his 1951 "Two Dogmas of Empiricism" that such a distinction is untenable.

Beside this, **Which statement is not enough for a proof?**

As a response to this: Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work.

People also ask, **What is the longest mathematical proof?** This is a list of unusually long mathematical proofs. Such proofs often use computational proof methods and may be considered non-surveyable . 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.

One may also ask, **What is mathematical proof?** Answer to this: 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.

**Are mathematical proofs analytic or synthetic?** The response is: A classic question in philosophy asks whether mathematical proofs are analytic or **synthetic**. Kant, who introduced the analytic–synthetic distinction, believed mathematical proofs are synthetic, whereas Quine argued in his 1951 "Two Dogmas of Empiricism" that such a distinction is untenable.

**Which statement is not enough for a proof?** Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work.