Fermat’s Last Theorem

Everything You Should Know About Fermat’s Last Theorem 

Fermat's Last Theorem is one of the most famous and celebrated results in mathematics. It was first posed by Pierre de Fermat in the 17th century and remained unsolved for centuries until it was finally proved by Andrew Wiles in 1994.

The statement of Fermat’s Last Theorem is simple: no three positive integers such as a, b and c can be satisfying the equations of aⁿ + bⁿ = cⁿ for any integers value of n bigger than 2. In other words, it is impossible to find any such combination of whole numbers that will produce a perfect power when each number is raised to a power greater than 2.

Looking at the initial values of n as 1 and 2, we get:

a + b = c                              and                                      a² + b² = c²

Note that the first equation is just a linear equation with infinite solutions.
The second equation may be easily recognizable as the Pythagoras Theorem where sum of the squares of perpendicular sides a and b of a right triangle add to the square of the hypotenuse c. This equation again has infinite solutions.

For many years after Fermat first proposed his theorem, mathematicians searched unsuccessfully for a proof. In the 19^th century, two important developments occurred which helped to finally solve it. Firstly, Niels Henrik Abel showed in 1824 that no such solution exists if n is greater than 5. Then in the 1880s, Sophus Lie used a new area of mathematics called group theory to prove that there are no solutions for any value of n greater than 4.

In 1993, Andrew Wiles announced his proof of Fermat's Last Theorem following over seven years of intense work on the problem. His proof relied on a technique known as modularity lifting and drew on ideas from number theory and algebraic geometry. It was accepted by the mathematical community and Wiles received numerous awards and accolades for his achievement.

Fermat's Last Theorem is a remarkable result which has been studied by mathematicians for centuries. Its proof was considered an impossible challenge, until Andrew Wiles finally solved the problem in 1994. It remains one of the most famous and celebrated results in mathematics today.