# Understanding the Fermat’s Last Theorem

The Fermat’s Last Theorem is one of the hardest problems in the history of mathematics.  The problem was written by Pierre de Fermat in 1637, and it was only solved more than 300 years later —  in 1995 by Professor Andrew Wiles.

But what is exactly the Fermat’s Last Theorem?

The Fermat’s Last Theorem is an extension of the Pythagorean Theorem.  Recall that the Pythagorean Theorem states that given a right triangle whose side lengths are $x, y$ and hypotenuse $z$, $x^2 + y^2 = z^2$ is satisfied. For example, a right triangle with side lengths $2$, and $3$ has hypotenuse $\sqrt{13}$.

There are some interesting things that we can see if we examine the side lengths of right triangles.  For instance, if we let the triples $(x,y,z)$ be the side lengths of a right triangle, where $z$ is the hypotenuse, we can find triples such that all lengths are integers. The triples  $(3, 4, 5)$, $(5,12,13)$, $(8,15,17)$ are integer triples, and they satisfy the Pythagorean Theorem. These triples are called Pythagorean Triples. It is not also difficult to see that there are infinitely many Pythagorean Triples (Can you see why?).

As number theory developed, mathematicians observed that there are no positive integer triples that satisfy the equation above given larger integer exponents.  For example, Fermat himself proved that there are no triples satisfying $x^4 + y^4 = z^4$.  Leonard Euler also proved that there are no integer triples satisfying $x^3 + y^3= z^3$.  Later in the 18th and 19th century, mathematicians proved that no such triples exist satisfying the equation for integral exponents up to four thousand. In fact, before these developments,  Fermat thought that he proved that

*No positive integers $x, y$, and $z$ can satisfy the equation $x^n + y^n = z^n$ for all integers n greater than 2.

In one of his notebooks, he wrote that

I have discovered a truly marvelous proof that it is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. This margin is too narrow to contain it.

believing that he actually proved the then conjecture.

Although Fermat’s proof was never found, mathematicians believe that his proof was incorrect because the mathematics of his time was not enough to prove the theorem.

In 1995, Andrew Wiles shocked the mathematics world by proving  The Fermat’s Last Theorem.  He spent 7 years (plus more than a year of revision, since his first proof was flawed) in his attic before he finally solved the problem.

*This is called the Fermat’s Last Theorem.

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