Fermat’s Last Theorem: A Mathematical Mystery

Pierre de Fermat

Fermat’s Last Theorem is one of the most famous and intriguing problems in the history of mathematics. It states as follows,

Theorem (Fermat's Last Theorem )
No three positive integers `x, y,` and `z` can satisfy the equation `x^n + y^n = z^n` for any integer n greater than 2.

For example, there are no positive integers x, y, and z such that `x^3 + y^3 = z^3` (the sum of two cubes is not a cube).

This simple-looking equation has fascinated mathematicians for centuries. It was first stated by Pierre de Fermat, a French lawyer and amateur mathematician, around 1637 in the margin of a copy of Arithmetica by Diophantus of Alexandria, an ancient Greek algebra book. Fermat wrote:

“It is impossible for a cube to be a sum of two cubes, a fourth power to be a sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly remarkable proof [of this theorem], but this margin is too small to contain it.”

1621 edition of the Arithmetica
 on which the right is the margin
that was too small to contain
 Fermat’s alleged proof

Unfortunately, Fermat never wrote down his proof anywhere else, and he died in 1665 without revealing it to anyone. His claim was discovered some 30 years later by his son Samuel, who published his father’s notes posthumously.

Fermat’s Last Theorem became one of the greatest challenges for mathematicians ever since. Many tried to find a proof or a counterexample (a set of numbers that would violate the theorem), but none succeeded. Some proofs were found for specific values of n (such as n = 4 by Fermat himself), but none were general enough to cover all cases.

The problem also stimulated the development of new branches of mathematics such as algebraic number theory and elliptic curves. It was also connected to other important conjectures such as the Taniyama-Shimura-Weil conjecture (also known as the modularity theorem), which relates elliptic curves to modular forms.

The breakthrough came in 1994 when Andrew Wiles, an English mathematician who had been fascinated by Fermat’s Last Theorem since he was 10 years old, announced that he had found a proof after seven years of secret work. His proof relied on some advanced techniques from algebraic geometry and number theory that were not available in Fermat’s time.

Sir Andrew John Wiles

However, his proof contained an error that was discovered by another mathematician during peer review. Wiles spent another year trying to fix his proof with the help of his former student Richard Taylor. Finally, in 1995 they published their corrected proof in Annals of Mathematics.

Wiles’ proof was hailed as one of the greatest achievements in mathematics and he received many honors and awards for it such as the Abel Prize (the Nobel Prize equivalent for mathematics) in 2016.

Fermat’s Last Theorem is now officially proven after more than 350 years since it was first stated by Fermat. However, many questions remain unanswered such as:

• Did Fermat really have a valid proof or did he make a mistake?
• If he had a proof, what was it like? Was it simple or complicated? Did it use methods known at his time or did he invent new ones?
• Is there another simpler or more elegant proof than Wiles’ one?
• Are there any applications or implications of Fermat’s Last Theorem for other areas of mathematics or science?

These questions may never be answered definitively but they will continue to inspire curiosity and creativity among mathematicians and non-mathematicians alike.